What is known about analytic continuation of the following product ($\forall p:u_p\in\mathbb{C}$ with $|u_p |\leq 1$):
$\prod\limits_{p\in\mathbb{P}} \left( 1-u_p p^{-s} \right)^{-1}$
Clearly, I am not asking this in full generality. Every method that works for an example $\{u_p\}$ is welcome.
The only reference I could find is: RYAVEC: THE ANALYTIC CONTINUATION OF EULER PRODUCTS WITH APPLICATIONS TO ASYMPTOTIC FORMULAE. Central here is the Lemma on page 612:
Lemma: Let $c$ and $z$ be complex numbers satisfying $|cz| < 1$ and $|z| < 1$. Then there exist complex numbers $h(m,c)$, $m = 1, 2,\ldots$ so that the equation
$\left( 1- c z \right)^{-1} = \prod\limits_{m=1}^\infty \left(\frac{1+z^m}{1-z^m}\right)^{h(m,c)}$ holds.
This then is used to express products like $\prod\limits_{p\in\mathbb{P}} \left( 1-u_p p^{-s} \right)^{-1}$ in terms of $\zeta(ms)$ (Riemann $\zeta$-function). Any other known attempts?
