In $\textit{Elliptic Fuctions according to Eisenstein and Kronecker}$, chap VIII, section 13 by A.Weil there is the following problem
For any integer $k \geq 0$ and $z, w \in \mathbb{C}$, the function $s \mapsto$ $K_k(z, w, s)$ extends as a meromorphic function to $\mathbb{C}$ with possible poles only at $s=0$ (if $k=0$ and $z \in L$ ) with residue $-\chi(w, z)$ or $s=1$ (if $k=0$ and $w \in L$ ) with residue $A(L)^{-1}$. Further, the functional equation $$ \Gamma(s) K_k(z, w, s)=A(L)^{k+1-2 s} \Gamma(k+1-s) K_k(w, z, k+1-s) \chi(w, z) $$ holds. The function $$ K_1(0,0, s)=\sum_{0 \neq \omega \in L} \frac{1}{|\omega|^{2 s}} $$ has a pole at $s=1$ with residue $A(L)^{-1}$.
Can someone help me to understand how this statements hold true?