Where are the zeros of $\prod\limits_p (1-(p-1)^z)$?

Question

Define $f(z)$ as the analytic continuation of $\prod\limits_p (1-(p-1)^z)$ where $z$ is complex and the product is over the odd primes $p$. Where are the zeros ($f(z)=0$) of this function ?

Answer 1

There is no a priori reason why this function should have an analytic continuation at all, beyond the region of convergence of the product.

Analytic continuation results are extremely difficult. For instance, the analytic continuation of the $L$-function of an elliptic curve $E/\mathbf Q$, conjectured by Hasse, was only proven as a corollary of the modularity theorem of Wiles and others, which is the main ingredient in the proof of Fermat's Last Theorem. This $L$-function is defined as an infinite product, whose terms are individually well understood - but the analytic continuation of the $L$-function to the whole complex plane is a very deep and highly nontrivial result.