Formula:DLMF:25.2:E12

<< Formula:DLMF:25.2:E11

formula in Definition and Expansions

Formula:DLMF:25.4:E1 >>

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{(2\pi)^{s}{\mathrm{e}^{-s-(\gamma s/2)}}}{2(s-1)\Gamma\left(\tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{\rho}\right){\mathrm{e}^{s/\rho}}}}}$

Constraint(s)

product over zeros ${\displaystyle{\displaystyle{\displaystyle\rho}}}$ of ${\displaystyle{\displaystyle{\displaystyle\RiemannZeta}}}$ with ${\displaystyle{\displaystyle{\displaystyle\Re{\rho}>0}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ : Euler's constant : http://dlmf.nist.gov/5.2#E3
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (12), Section 25.2 of DLMF.

URL links

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