I know that the summation 1/p^s (s>1 and p prime) converges. Now, I define J(s) = summation 1/p^s. Are there any "well known" values for J(2),J(3),J(4), etc? like we all know that zeta(2)= (pi^2)/6, zeta(4)= (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


