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.

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 handwaving/heuristic arguments for the difficulty (not sure how good/convincing they are):
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 MeisselMertens constant, which is approxiamtely $0.2614972$. 

