Epstein zeta function
= Epstein zeta function
{c}
{title2=$E_\Lambda(s)$}
{wiki}
For a full-rank <Euclidean lattice> in $\mathbb R^n$, its Epstein zeta function is $E_\Lambda(s)=\sum_{0\ne x\in\Lambda}|x|^{-2s}$, initially convergent for $\operatorname{Re}s>n/2$. A <Mellin transform> of its <theta function> gives its <completed Epstein zeta function> and a meromorphic continuation. Its sole pole has residue $\pi^{n/2}/[m(\Lambda)\Gamma(n/2)]$ at $s=n/2$.