Riemann Zeta Function Curiosity?

Alright, so I was experimenting with my TI-89, and I found out that the calculator had memorized the solution to ζ(2) - nameley, when I entered the summation formula explicity as the sum from k = 1 to infinity of 1 / k^2, it gave me the exact answer π^2/6.

Further experimentation yielded ζ(4) = π^4/90, ζ(6) = π^6/945, and onwards up the trail until I got bored, unable to recognize any elegant pattern.

My question is, is there an explicit formula for ζ(n) for all positive even integers n? And will this always yield an answer in the form c*π^n, where c is a rational number? (c is in the form 1 / d for the first five even integers, but once you hit n = 12 that nice hypothesis goes out the window. ζ(12) = 691 π^12 / 638512875)

Insights, proofs, explanations - all are welcome. I'd like to see where this leads.