Numerical evidence suggests that:

$$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$

with $p_n$ is the n-th prime number, converges for all $\Re(s) > \frac12$.

Note that for $s=1$ the function reduces to the well known limit for $e^{-\gamma}$ (see here).

Could this be proven?

`s=1/2+10.0^(-8)`

and`v=10^n`

as $n$ grows? I get steady increase, for $n=8$ get $38.3...$, while for n=6 it is 28.72...? If it diverges very slowly like log(log(N)) you wouldn't detect it this way AFAICT. Btw, you may have precision issues, not sure. $\endgroup$