Epstein zeta function ($E_{\Lambda }(s)$)

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 $\mathrm{Re}s>n/2$. 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 $s=n/2$.

The completion ${\mathcal{E}}_{\Lambda }(s)={\pi }^{-s}\Gamma (s)E_{\Lambda }(s)$ satisfies ${\mathcal{E}}_{\Lambda }(s)=m(\Lambda {)}^{-1}{\mathcal{E}}_{{\Lambda }^{\vee }}(n/2-s)$. Its simple poles at zero and $n/2$ have residues $-1$ and $m(\Lambda {)}^{-1}$. The pole-subtracted theta integral for an Epstein zeta function gives both the continuation and this dual lattice symmetry.

Put $a=n/2$, $m=m(\Lambda )$ and $A_{\Lambda }(s)={\int }_1^{\infty }({\Theta }_{\Lambda }(it)-1)t^{s-1}dt$, which is entire. Splitting the Mellin transform at one and using the lattice theta functional equation gives ${\mathcal{E}}_{\Lambda }(s)=A_{\Lambda }(s)+m^{-1}{A}_{{\Lambda }^{\vee }}(a-s)+m^{-1}/(s-a)-1/s$. The two rational terms explicitly retain the contributions of the zero lattice vector.