An explicit expression of the Lerch zeta function on maximal domains of holomorphy

We give two results on the Lerch zeta function $\mathrm{\Phi }(z,s,w)$. The first is to give an explicit expression providing both the analytic continuation of $\mathrm{\Phi }$ in $n$-variables $(n\in \{1,2,3\})$ to maximal domains of holomorphy in ${\mathbb{C}}^n$ with computable evaluation and an extended formula for the special values of $\mathrm{\Phi }$ at non-positive integers in the variable $s$. The second is to show that Lerch’s functional equation is essentially the same as Apostol’s functional equation using the first result.