We can solve for zeta(s) for all values (with the only pole being at z = 1) via analytical continuation. This process extends the traditional definition (which only applies when Re(z) > 1 and expands it to Re(z) < 0 and then further continuation gives definition into the strip between them.
However, I assume you're discussing the Riemann hypothesis, which states that all zeroes of this function occur at either s = -2 * n (where n in mathbb N; these are called the trivial zeroes) or at a point in the complex plane with s = 1/2 + bi where b in mathbb R. This is the major 'unproven' part of zeta(s).
