Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$) I know that $\sum_p p^{-s}$, $s>1$, converges. Now, I define $J(s) = \sum_p p^{-s}$. Are there any "well known" values for $J(2)$, $J(3)$, $J(4)$, etc? We all know that $\zeta(2)= \frac{\pi^2}{6}$, $\zeta(4)=\frac{\pi^4}{90}$, etc.
 A: No, in the sense that there are (for all I know) no identies along the lines of those for $\zeta(s)$, that you recalled, known.  
Your function $J$ is sometimes called the prime zeta function.
You can find some information, some approciamte numerical values, plots, and pointers to the literature, e.g., at
http://mathworld.wolfram.com/PrimeZetaFunction.html
and 
http://en.wikipedia.org/wiki/Prime_zeta_function
Two related hand-waving/heuristic arguments for the difficulty (not sure how good/convincing they are):


*

*The values would 'encode' quite precise information on the set of primes.

*The arithmetic function you are summing, that is, $f(n) = n^{-s}$ if $n$ is prime, and $f(n)=0$ if $n$ is not prime, is not a 'nice' arithmetic function; for example it is not multiplicative.
A related note that might interest you, in case you are not aware of it:
As you say $\sum_p p^{-1}$ diverges. However, the rate of divergence is fairly precisely known.
Namely, by Mertens's Second Theorem
$$\lim_{n \to \infty} \left ( \sum_{p\le n} p^{-1}\right ) - \log \log n $$
exists, and is equal to (or perhaps, rather defines) the  Meissel--Mertens constant,
which is approxiamtely $0.2614972$.
