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)#.
