About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$

From the answers and feedback I got, we know that an analytic continuation exists for $0<s<1$ and there is no analytic continuation for $s>1.$

What's the maximal analytic continuation of $\varphi(s)?$

Doing this will help me better understand how the function behaves.