Corollary 3.5 (Zeroes and poles of $zeta$). The $\zeta$ function extends to a meromorphic function in $ℂ$ and it has

• (i)
  Only one pole, which is a simple pole at $s=1$, residue $1$.
• (ii)
  Simple zeroes at $s=-2,-4,-6,\dots$.
• (iii)
  Any other zeroes satisfy $0\le \mathrm{Re}(s)\le 1$.