The prime zeta function is defined as
$$P(s)=\sum_{p\,\in\mathrm{\mathcal P}} \frac{1}{p^s},$$
where $\mathcal P$ is the set of prime numbers. It converges for all $\Re(s)>1$.
There is a related function defined as
$$\int_s^\infty P(t)\,dt = \sum_p \frac{1}{p^s\log p}.$$
Question. Are there any known closed-form value of $P(s)$ or $\int_s^\infty P(t)\,dt$?
There is no closed form for the prime zeta function, but it can be computed as $$ P(s)=\sum_{n\ge1} \mu(n)\frac{\log\zeta(ns)}{n} $$ which is more efficient than the standard form. In some sense these are adding in different directions, and a more efficient method can be obtained by adding in both directions and subtracting the overlap.