# Formula:DLMF:25.5:E20

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma% \left(1-s\right)}{2\cpi\iunit}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{\expe^{-z}-1% }\diffd z}}}$

## Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}}$ &

The integration contour is a loop around the negative real axis; it starts at ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$, encircles the origin once in the positive direction without enclosing any of the points

${\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 4\pi i,\ldots,}}}$ and returns to ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

& : logical and
: Riemann zeta function : http://dlmf.nist.gov/25.2#E1
: Euler's gamma function : http://dlmf.nist.gov/5.2#E1
: ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
: integral : http://dlmf.nist.gov/1.4#iv
: the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
: differential : http://dlmf.nist.gov/1.4#iv