# Formula:DLMF:25.11:E30

${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)% }{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}\frac{{\mathrm{e}^{az}}z^{s-1}}{1-{% \mathrm{e}^{z}}}\mathrm{d}z}}$

## Constraint(s)

${\displaystyle{\displaystyle s\neq 1}}$ &
${\displaystyle{\displaystyle\Re{a}>0}}$ &

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

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

## 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
: Hurwitz zeta function : http://dlmf.nist.gov/25.11#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
: real part : http://dlmf.nist.gov/1.9#E2