Gamma factors (reviewed)

The complex functions \[ \Gamma_\R(s) := \pi^{-s/2}\Gamma(s/2)\qquad\text{and}\qquad \Gamma_\C(s):= 2(2\pi)^{-s}\Gamma(s) \] that appear in the functional equation of an L-function are known as gamma factors. Here $\Gamma(s):=\int_0^\infty e^{-t}t^{s-1}dt$ is Euler's gamma function.

The gamma factors satisfy $ \Gamma_\C(s) = \Gamma_\R(s) \Gamma_\R(s + 1) $ and can also be viewed as “missing” factors of the Euler product of an L-function corresponding to (real or complex) archimedean places.