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

We give two results on the Lerch zeta function $\Phi(z,\,s,\,w)$. The first is to give an explicit expression providing both the analytic continuation of $\Phi$ 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 $\Phi$ 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.