To clarify, I'm asking about series where $n$th term depends only on $n$th prime number. From the famous result (by Euler, I think) we have:
$$\sum_{n=1}^\infty \ln \left(1-\frac{1}{p_n^s} \right)=-\ln \zeta(s)$$
As a particular case:
$$\sum_{n=1}^\infty \ln \left(1-\frac{1}{p_n^2} \right)=-\ln \frac{\pi^2}{6}$$
What about the general series:
$$\sum_{n=1}^\infty f(p_n)$$
Does it have a nontrivial closed form for some other function $f$?
In general, I'm also interested in infinite products (they can be easily written as series, as I did for the titular one), nested radicals, continued fractions, etc.
Edit
Thanks to the great advice of @user1952009, I looked up Dirichlet beta function, which has the prime expansion:
$$\sum_{n=2}^\infty \ln \left(1-\frac{(-1)^{(p_n-1)/2}}{p_n^s} \right)=-\ln \beta(s)$$
Among it's special values we have:
$$\sum_{n=2}^\infty \ln \left(1-\frac{(-1)^{(p_n-1)/2}}{p_n} \right)=-\ln \frac{\pi}{4}$$
$$\sum_{n=2}^\infty \ln \left(1-\frac{(-1)^{(p_n-1)/2}}{p_n^2} \right)=-\ln G$$
(Catalan's constant)
$$\sum_{n=2}^\infty \ln \left(1-\frac{(-1)^{(p_n-1)/2}}{p_n^3} \right)=-\ln \frac{\pi^3}{32}$$
Etc.