Convergence and analytic continuation of a specific infinite product I've been working on this for months, and I think it's time to give up and appeal to the internet for help.
$$\prod_{p}p^{\large 1/\left(\,1 -p^{s}\,\right)}$$ 
When does this converge and how can I get a continuation of it over all
$s\in \mathbb{C}$. ( $p$ means over the primes ).
Don't be afraid to get too technical on me. Some of complex analysis is a bit beyond me, but if it leads to an answer to my problem, I'll learn any bit of it that you think could help me. Really, any help would be appreciated. Thank you in advance.

The given product equals
$$ \exp\left(-\sum_p \frac{\log p}{p^s-1}\right)\tag{1}$$
that is absolutely convergent for any $s:\text{Re}(s)>1$. On the same region Euler's product
$$ \zeta(s)=\prod_{p}\left(1-\frac{1}{p^s}\right)^{-1} \tag{2} $$
leads to
$$ \frac{d}{ds}\log\zeta(s) = \frac{\zeta'(s)}{\zeta(s)} = -\sum_{p}\frac{\log p}{p^s-1}\tag{3} $$
and to:
$$ \prod_p  p^{\frac{1}{p^s-1}}=\exp\left(\frac{\zeta'(s)}{\zeta(s)}\right)=\exp\left(-\sum_{n\geq 1}\frac{\Lambda(n)}{n^s}\right)\tag{4} $$
where $\Lambda(n)$ is Von Mangoldt function.The analytic continuation to the complex plane can be derived from the analytic continuation for the $\zeta$ function.
