Formula:DLMF:25.15:E3

<< Formula:DLMF:25.15:E2

formula in Dirichlet L-functions

Formula:DLMF:25.15:E4 >>

${\displaystyle{\displaystyle{\displaystyle L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\HurwitzZeta@{s}{\frac{r}{k}}}}}$

Constraint(s)

hold for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ if ${\displaystyle{\displaystyle{\displaystyle\chi\neq\chi_{1}}}}$, and for all ${\displaystyle{\displaystyle{\displaystyle s}}}$ (${\displaystyle{\displaystyle{\displaystyle\neq 1}}}$) if ${\displaystyle{\displaystyle{\displaystyle\chi=\chi_{1}}}}$

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

${\displaystyle{\displaystyle{\displaystyle L}}}$ : Dirichlet ${\displaystyle{\displaystyle{\displaystyle L}}}$-function : http://dlmf.nist.gov/25.15#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1

Bibliography

Equation (3), Section 25.15 of DLMF.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:DLMF:25.15:E2

formula in Dirichlet L-functions

Formula:DLMF:25.15:E4 >>