In Riemann's paper on page 3 we see:
$\Gamma(\frac{s}{2} - 1)\pi^{-\frac{s}{2}}\zeta(s) = \frac{1}{s(s-1)} + \int\limits_1^\infty\psi(x)\left(x^{\frac{s}{2} - 1} + x^{-\frac{1+s}{2}}\right)dx$
In the above expression $\zeta(s) = \sum \frac{1}{n^s}$ which converges only for $s > 1$
Riemann goes on to define $\xi(s) = \Gamma(\frac{s}{2})(s-1)\pi^{-\frac{s}{2}}\zeta(s)$
On page 4, he is talking about the zeros of $\xi(s)$.
Here is my question: The $\zeta(s)$ in $\xi(s)$ converges only for $s > 1$ and has no zeros anywhere. It is possible the RHS has zeros. But I am not seeing how this implies $\zeta(s)$ which was written in the top of page 3 as $\sum\frac{1}{n^s}$ has zeros.
Obviously I am missing something here.
Thank you