DLMF:25.5.E13 (Q7626)

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DLMF:25.5.E13
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    Statements

    Defining formula
    ζ ⁡ ( s ) = π s / 2 s ⁢ ( s - 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ s ) + π s / 2 Γ ⁡ ( 1 2 ⁢ s ) ⁢ ∫ 1 ∞ ( x s / 2 + x ( 1 - s ) / 2 ) ⁢ ω ⁢ ( x ) x ⁢ d x , Riemann-zeta 𝑠 superscript 𝜋 𝑠 2 𝑠 𝑠 1 Euler-Gamma 1 2 𝑠 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 superscript subscript 1 superscript 𝑥 𝑠 2 superscript 𝑥 1 𝑠 2 𝜔 𝑥 𝑥 𝑥 {\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,}}
    0 references
    DLMF id
    DLMF:25.5.E13
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    constraint
    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aldec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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