Pole-subtracted theta integral for an Epstein zeta function (${\mathcal{E}}_{\Lambda }(s)=A_{\Lambda }(s)+m^{-1}{A}_{{\Lambda }^{\vee }}(n/2-s)+m^{-1}/(s-n/2)-1/s$)

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.