Consider the following function :

$$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$

Brun's theorem tells us that $F(1)$ is finite.

We are looking for the infiniteness of $F(0)$. Indeed, this is the Twin Prime Conjecture, supported by extensive numerical evidence.

Is there a numerical estimate for a value of $s\in(0,1)$ such that $F(s)$ seems to diverge for that value (and all values less than that of course)?