Let $P$ be the set of all odd prime numbers. I am looking for all $s\in(1,\infty)$ for them

$ A=\prod_{p\in P} (1+\frac{1}{(p-1)^s})^{p-1} $

exists (i.e. is finite). I know that it should be somehow related to Riemann zeta function but I was not sure how can I pursue the calculations.

If I use natural logarithm I will get:

$ \ln(A)=\sum_{p\in P} (p-1) \ln(1+ \frac{1}{(p-1)^s})$

whcih I am not sure is useful of not!