#analytic-number-theory

The Riemann zeta function is defined by $\zeta (s)={\sum }_{n\ge 1}{n}^{-s}$ for $\mathrm{Re}(s)>1$ and extended by analytic continuation to all complex $s\ne 1$. It vanishes at $s=-2,-4,-6,\dots$. Prove that every other zero of $\zeta$ has real part exactly $1/2$.