DLMF:20.10.E2 (Q6838)

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DLMF:20.10.E2
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    ∫ 0 ∞ x s - 1 ⁢ ( θ 3 ⁡ ( 0 | i ⁢ x 2 ) - 1 ) ⁢ d x = π - s / 2 ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ ζ ⁡ ( s ) , superscript subscript 0 superscript 𝑥 𝑠 1 Jacobi-theta-tau 3 0 𝑖 superscript 𝑥 2 1 𝑥 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|% ix^{2}\right)-1)\mathrm{d}x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta% \left(s\right),}}
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    DLMF:20.10.E2
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