the zeta functional equation

The zeta function hides a profound symmetry that swaps the point s with its reflection 1 - s across the line Re s = 1/2. Riemann's functional equation makes it precise. In its symmetric form, define the completed zeta xi(s) = pi^(-s/2) Gamma(s/2) zeta(s); then xi(s) = xi(1 - s). The factor pi^(-s/2) Gamma(s/2) is exactly the gamma 'fudge factor' (the local factor at infinity) that makes the equation clean. In its unsymmetric form it reads zeta(s) = 2^s pi^(s-1) sin(pi s/2) Gamma(1 - s) zeta(1 - s).

Where does such a symmetry come from? Riemann's proof runs through a theta function, theta(t) = sum over n of e^(-pi n^2 t), which satisfies its own beautiful transformation theta(1/t) = sqrt(t) theta(t) (a modular relation). Feeding theta into a Mellin (gamma-type) integral produces the completed zeta xi(s), and the theta transformation t -> 1/t becomes precisely the reflection s -> 1 - s. So the s-to-(1-s) symmetry of zeta is the analytic shadow of the t-to-(1/t) symmetry of a theta function — special functions of analysis tying number theory to the geometry of lattices.

The functional equation does real work. It instantly computes zeta in the left half-plane from values in the right: the trivial zeros at s = -2, -4, -6, ... appear because sin(pi s/2) vanishes there, and special values like zeta(-1) = -1/12 fall out. It also makes the line Re s = 1/2 the axis of symmetry, which is why that line is 'critical'. A caveat: the gamma factor is not optional decoration — without pi^(-s/2) Gamma(s/2) the symmetry is not s <-> 1-s, and the poles of that gamma factor are precisely what conspire with zeta's pole and trivial zeros to keep xi(s) entire (apart from two simple poles).