3 Functional Equation
3.1 Completed Riemann Zeta Function
\(\xi (s) = \tfrac {1}{2} s(s-1) \pi ^{-s/2} \Gamma (s/2) \zeta (s)\).
For any Schwartz-Bruhat function \(\Phi \) on \(\mathbb {A}\): \(Z(\Phi , s) = Z(\hat\Phi , 1-s)\). Follows from Haar self-duality + Poisson summation over \(\mathbb {Q} \subset \mathbb {A}\).
\(\xi (s) = \xi (1-s)\) for all \(s \in \mathbb {C}\) away from poles. This is the arithmetic shadow symmetry.
If \(\xi (\rho ) = 0\) then \(\xi (1-\rho ) = 0\).
\(s = 1-s \iff \mathrm{Re}(s) = \tfrac {1}{2}\).