Let \(\zeta\) be the Riemann zeta function, continued meromorphically to \(\CC\). The only zeros of \(\zeta\) outside the critical strip \(\ts{s\st0\le\Re s\le1}\) are the simple zeros at \(s=-2,-4,-6,\ldots\). Moreover, \(\zeta\) has no zeros on the line \(\Re s=1\).
For \(\Re s>1\) the Euler product \(\zeta(s)=\prod_p(1-p^{-s})^{-1}\) converges absolutely, and an absolutely convergent product of nonzero factors is nonzero. For \(\Re s<0\) use the functional equation \[ \zeta(s)=\pi^{s-1/2}\,\frac{\Gamma\big(\frac{1-s}{2}\big)}{\Gamma\big(\frac s2\big)}\,\zeta(1-s). \] There \(\Re(1-s)>1\), so \(\zeta(1-s)\neq0\); \(\Gamma\big(\frac{1-s}{2}\big)\) is finite and nonzero because \(\Gamma\) has no zeros and its poles lie at \(0,-1,-2,\ldots\); and \(1/\Gamma(s/2)\) is entire with simple zeros exactly at \(s=0,-2,-4,\ldots\). So the zeros of \(\zeta\) with \(\Re s<0\) are the simple zeros at \(s=-2,-4,\ldots\). The nonvanishing of \(\zeta\) on \(\Re s=1\) is the theorem of Hadamard and de la Vallée Poussin.
The zeros at the negative even integers are the trivial zeros of \(\zeta\). The Riemann hypothesis is the conjecture that every zero of \(\zeta\) in the critical strip lies on the line \(\Re s=\frac12\).