25.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.5.E1	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\diff{x}	${\displaystyle{\displaystyle\Re s>1,\Re s>0}}$	Zeta(s) = (1)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- 1), x = 0..infinity)	Zeta[s] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- 1], {x, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
25.5.E2	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\diff{x}	${\displaystyle{\displaystyle\Re s>1,\Re(s+1)>0}}$	Zeta(s) = (1)/(GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)- 1)^(2)), x = 0..infinity)	Zeta[s] == Divide[1,Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]- 1)^(2)], {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 2]	Successful [Tested: 2]
25.5.E3	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\diff{x}	${\displaystyle{\displaystyle\Re s>0}}$	Zeta(s) = (1)/((1 - (2)^(1 - s))*GAMMA(s))*int(((x)^(s - 1))/(exp(x)+ 1), x = 0..infinity)	Zeta[s] == Divide[1,(1 - (2)^(1 - s))*Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]+ 1], {x, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
25.5.E4	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\diff{x}	${\displaystyle{\displaystyle\Re s>0,\Re(s+1)>0}}$	Zeta(s) = (1)/((1 - (2)^(1 - s))*GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)+ 1)^(2)), x = 0..infinity)	Zeta[s] == Divide[1,(1 - (2)^(1 - s))*Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]+ 1)^(2)], {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 3]	Successful [Tested: 3]
25.5.E5	${\displaystyle{\displaystyle\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{x^{s+1}}\mathrm{d}x}}$ \Riemannzeta@{s} = -s\int_{0}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}}\diff{x}	${\displaystyle{\displaystyle-1<\Re s,\Re s<0}}$	Zeta(s) = - s*int((x - floor(x)-(1)/(2))/((x)^(s + 1)), x = 0..infinity)	Zeta[s] == - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x)^(s + 1)], {x, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Failed [1 / 1] Result: Complex[-2500601.4984594644, 2.5458720000534374*^-17] Test Values: {Rule[s, -0.5]}
25.5.E6	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\diff{x}	${\displaystyle{\displaystyle\Re s>-1,\Re s>0}}$	Zeta(s) = (1)/(2)+(1)/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(x)), x = 0..infinity)	Zeta[s] == Divide[1,2]+Divide[1,s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 3]	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[s, 1.5]} Result: Indeterminate Test Values: {Rule[s, 0.5]} ... skip entries to safe data
25.5.E7	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{\BernoullinumberB{2m}}{(2m)!}\Pochhammersym{s}{2m-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{\BernoullinumberB{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\diff{x}	${\displaystyle{\displaystyle\Re s>-(2n+1),\Re s>0}}$	Zeta(s) = (1)/(2)+(1)/(s - 1)+ sum((bernoulli(2*m))/(factorial(2*m))*pochhammer(s, 2*m - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*m))/(factorial(2*m))*(x)^(2*m - 1), m = 1..n))*((x)^(s - 1))/(exp(x)), x = 0..infinity), m = 1..n)	Zeta[s] == Divide[1,2]+Divide[1,s - 1]+ Sum[Divide[BernoulliB[2*m],(2*m)!]*Pochhammer[s, 2*m - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*m],(2*m)!]*(x)^(2*m - 1), {m, 1, n}, GenerateConditions->None])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}, GenerateConditions->None], {m, 1, n}, GenerateConditions->None]	Aborted	Aborted	Failed [2 / 3] Result: .1027534444e-1 Test Values: {s = 3/2, n = 3} Result: .24417579e-1 Test Values: {s = 2, n = 3}	Skipped - Because timed out
25.5.E8	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2(1-2^{-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh x}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{2(1-2^{-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh@@{x}}\diff{x}	${\displaystyle{\displaystyle\Re s>1,\Re s>0}}$	Zeta(s) = (1)/(2*(1 - (2)^(- s))*GAMMA(s))*int(((x)^(s - 1))/(sinh(x)), x = 0..infinity)	Zeta[s] == Divide[1,2*(1 - (2)^(- s))*Gamma[s]]*Integrate[Divide[(x)^(s - 1),Sinh[x]], {x, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
25.5.E9	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{x^{s}}{(\sinh x)^{2}}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{2^{s-1}}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{x^{s}}{(\sinh@@{x})^{2}}\diff{x}	${\displaystyle{\displaystyle\Re s>1,\Re(s+1)>0}}$	Zeta(s) = ((2)^(s - 1))/(GAMMA(s + 1))*int(((x)^(s))/((sinh(x))^(2)), x = 0..infinity)	Zeta[s] == Divide[(2)^(s - 1),Gamma[s + 1]]*Integrate[Divide[(x)^(s),(Sinh[x])^(2)], {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 2]	Skipped - Because timed out
25.5.E10	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\left(\frac{1}{2}\pi x\right)}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos@{s\atan@@{x}}}{(1+x^{2})^{s/2}\cosh@{\frac{1}{2}\pi x}}\diff{x}		Zeta(s) = ((2)^(s - 1))/(1 - (2)^(1 - s))*int((cos(s*arctan(x)))/((1 + (x)^(2))^(s/2)* cosh((1)/(2)*Pi*x)), x = 0..infinity)	Zeta[s] == Divide[(2)^(s - 1),1 - (2)^(1 - s)]*Integrate[Divide[Cos[s*ArcTan[x]],(1 + (x)^(2))^(s/2)* Cosh[Divide[1,2]*Pi*x]], {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 6]	Skipped - Because timed out
25.5.E11	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin@{s\atan@@{x}}}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}		Zeta(s) = (1)/(2)+(1)/(s - 1)+ 2*int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/2)*(exp(2*Pi*x)- 1)), x = 0..infinity)	Zeta[s] == Divide[1,2]+Divide[1,s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 6]	Successful [Tested: 6]
25.5.E12	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin@{s\atan@@{x}}}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\diff{x}		Zeta(s) = ((2)^(s - 1))/(s - 1)- (2)^(s)* int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/2)*(exp(Pi*x)+ 1)), x = 0..infinity)	Zeta[s] == Divide[(2)^(s - 1),s - 1]- (2)^(s)* Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/2)*(Exp[Pi*x]+ 1)], {x, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 6]	Successful [Tested: 6]
25.5.E13	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\pi^{s/2}}{s(s-1)\Gamma\left(\frac{1}{2}s\right)}+\frac{\pi^{s/2}}{\Gamma\left(\frac{1}{2}s\right)}\*\int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{\pi^{s/2}}{s(s-1)\EulerGamma@{\frac{1}{2}s}}+\frac{\pi^{s/2}}{\EulerGamma@{\frac{1}{2}s}}\*\int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\diff{x}	${\displaystyle{\displaystyle s\neq 1,\Re(\frac{1}{2}s)>0}}$	Zeta(s) = ((Pi)^(s/2))/(s*(s - 1)*GAMMA((1)/(2)*s))+((Pi)^(s/2))/(GAMMA((1)/(2)*s))* int(((x)^(s/2)+ (x)^((1 - s)/2))*(omega(x))/(x), x = 1..infinity)	Zeta[s] == Divide[(Pi)^(s/2),s*(s - 1)*Gamma[Divide[1,2]*s]]+Divide[(Pi)^(s/2),Gamma[Divide[1,2]*s]]* Integrate[((x)^(s/2)+ (x)^((1 - s)/2))*Divide[\[Omega][x],x], {x, 1, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [30 / 30] Result: Float(infinity)+Float(infinity)*I Test Values: {omega = 1/2*3^(1/2)+1/2*I, s = 3/2} Result: Float(infinity)+Float(infinity)*I Test Values: {omega = 1/2*3^(1/2)+1/2*I, s = 1/2} ... skip entries to safe data	Skipped - Because timed out
25.5.E14	${\displaystyle{\displaystyle\omega(x)\equiv\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2}\left(\theta_{3}\left(0\middle|ix\right)-1\right)}}$ \omega(x)\defeq\sum_{n=1}^{\infty}e^{-n^{2}\pi x} = \frac{1}{2}\left(\Jacobithetatau{3}@{0}{ix}-1\right)		omega(x) = sum(exp(- (n)^(2)* Pi*x), n = 1..infinity) = (1)/(2)*(JacobiTheta3(0,exp(I*Pi*I*x))- 1)	\[Omega][x] == Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}, GenerateConditions->None] == Divide[1,2]*(EllipticTheta[3, 0, Exp[I*Pi*(I*x)]]- 1)	Failure	Failure	Error	Failed [30 / 30] Result: Plus[-0.008983297533541545, False] Test Values: {Rule[x, 1.5], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-0.008983297533541545, False] Test Values: {Rule[x, 1.5], Rule[ω, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
25.5.E15	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi}\*\int_{0}^{\infty}(\ln\left(1+x\right)-\psi\left(1+x\right))x^{-s}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi}\*\int_{0}^{\infty}(\ln@{1+x}-\digamma@{1+x})x^{-s}\diff{x}		Zeta(s) = (1)/(s - 1)+(sin(Pi*s))/(Pi)* int((ln(1 + x)- Psi(1 + x))*(x)^(- s), x = 0..infinity)	Zeta[s] == Divide[1,s - 1]+Divide[Sin[Pi*s],Pi]* Integrate[(Log[1 + x]- PolyGamma[1 + x])*(x)^(- s), {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [5 / 6] Result: Float(-infinity) Test Values: {s = -3/2} Result: Float(infinity) Test Values: {s = 3/2} ... skip entries to safe data	Skipped - Because timed out
25.5.E16	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\psi'\left(1+x\right)\right)x^{1-s}\mathrm{d}x}}$ \Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\digamma'@{1+x}\right)x^{1-s}\diff{x}		Zeta(s) = (1)/(s - 1)+(sin(Pi*s))/(Pi*(s - 1))* int(((1)/(1 + x)- subs( temp=1 + x, diff( Psi(temp), temp$(1) ) ))*(x)^(1 - s), x = 0..infinity)	Zeta[s] == Divide[1,s - 1]+Divide[Sin[Pi*s],Pi*(s - 1)]* Integrate[(Divide[1,1 + x]- (D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x))*(x)^(1 - s), {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [4 / 6] Result: Float(-infinity) Test Values: {s = -3/2} Result: Float(infinity) Test Values: {s = -1/2} ... skip entries to safe data	Skipped - Because timed out
25.5.E17	${\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)}{\pi}\int_{0}^{\infty}\left(\gamma+\psi\left(1+x\right)\right)x^{-s-1}\mathrm{d}x}}$ \Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi}\int_{0}^{\infty}\left(\EulerConstant+\digamma@{1+x}\right)x^{-s-1}\diff{x}		Zeta(1 + s) = (sin(Pi*s))/(Pi)*int((gamma + Psi(1 + x))*(x)^(- s - 1), x = 0..infinity)	Zeta[1 + s] == Divide[Sin[Pi*s],Pi]*Integrate[(EulerGamma + PolyGamma[1 + x])*(x)^(- s - 1), {x, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [6 / 6] Result: Float(-infinity) Test Values: {s = -3/2} Result: Float(-infinity) Test Values: {s = 3/2} ... skip entries to safe data	Skipped - Because timed out
25.5.E18	${\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)}{\pi s}\int_{0}^{\infty}\psi'\left(1+x\right)x^{-s}\mathrm{d}x}}$ \Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi s}\int_{0}^{\infty}\digamma'@{1+x}x^{-s}\diff{x}		Zeta(1 + s) = (sin(Pi*s))/(Pi*s)*int(subs( temp=1 + x, diff( Psi(temp), temp$(1) ) )*(x)^(- s), x = 0..infinity)	Zeta[1 + s] == Divide[Sin[Pi*s],Pi*s]*Integrate[(D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [5 / 6] Result: Float(infinity) Test Values: {s = -3/2} Result: Float(infinity) Test Values: {s = 3/2} ... skip entries to safe data	Failed [5 / 6] Result: Complex[1.225514223589735984806690657246494542888138*^+10484, 2.5458720000534374*^-17] Test Values: {Rule[s, -1.5]} Result: 2.9996624622276097*^37 Test Values: {Rule[s, 1.5]} ... skip entries to safe data
25.5.E19	${\displaystyle{\displaystyle\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s\right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi^{(m)}}\left(1+x\right)x^{-s}\mathrm{d}x}}$ \Riemannzeta@{m+s} = (-1)^{m-1}\frac{\EulerGamma@{s}\sin@{\pi s}}{\pi\EulerGamma@{m+s}}\*\int_{0}^{\infty}\digamma^{(m)}@{1+x}x^{-s}\diff{x}	${\displaystyle{\displaystyle\Re s>0,\Re(m+s)>0}}$	Zeta(m + s) = (- 1)^(m - 1)*(GAMMA(s)*sin(Pi*s))/(Pi*GAMMA(m + s))* int(subs( temp=1 + x, diff( Psi(temp), temp$(m) ) )*(x)^(- s), x = 0..infinity)	Zeta[m + s] == (- 1)^(m - 1)*Divide[Gamma[s]*Sin[Pi*s],Pi*Gamma[m + s]]* Integrate[(D[PolyGamma[temp], {temp, m}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [2 / 3] Result: Float(infinity) Test Values: {s = 3/2, m = 3} Result: Float(-infinity) Test Values: {s = 2, m = 3}	Skipped - Because timed out
25.5.E20	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\mathrm{d}z}}$ \Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\diff{z}	${\displaystyle{\displaystyle\Re(1-s)>0}}$	Zeta(s) = (GAMMA(1 - s))/(2*Pi*I)*int(((z)^(s - 1))/(exp(- z)- 1), z = - infinity..(0 +))	Zeta[s] == Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[(z)^(s - 1),Exp[- z]- 1], {z, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
25.5.E21	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\mathrm{d}z}}$ \Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\diff{z}	${\displaystyle{\displaystyle\Re(1-s)>0}}$	Zeta(s) = (GAMMA(1 - s))/(2*Pi*I*(1 - (2)^(1 - s)))* int(((z)^(s - 1))/(exp(- z)+ 1), z = - infinity..(0 +))	Zeta[s] == Divide[Gamma[1 - s],2*Pi*I*(1 - (2)^(1 - s))]* Integrate[Divide[(z)^(s - 1),Exp[- z]+ 1], {z, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error