The sum $$ \sum\limits_{p,q \text{ primes } p \le q} \frac{1}{pq\log(pq)}$$

is related to a conjecture of Erdős about primitive sequences.

It converges because the sequence is primitive. If my calculations are correct a lower bound is $1.062$.

Is it known to what is equal the sum?

individualprimes and that is well beyond the current state of art. Actually, even if you can express things in terms of $\zeta$, it is rather questionable whether it can be called "an explicit formula" because we don't know much about the values of $\zeta$ except at the even integers. – fedja Jun 4 '12 at 14:48