Define $\varphi(s)=\prod_{n=1}^\infty \exp\big(n^{-s}\big)$ where $s=\exp a + i\exp b$ for $a,b \in \Bbb R.$
Does there exist an analytic continuation of $\varphi(s)?$
For real $s>1$ the product converges.
I'm wondering if there's an analytic continuation of $\varphi(s)$ and what it is in terms of familiar functions.
The space of numbers, $s$ forms a plane isomorphic to $\Bbb C.$