Functional equation (L-function)

In mathematics, the L-functions of number theory have certain functional equations, as one of their characteristic properties. There is an elaborate theory of what these should be; much of it still conjectural. For example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation from the infinite series definition. That is, writing as is conventional σ for the real part of s, the functional equation relates the cases

σ > 1 and σ < 0,

and also changes a case with

0 < σ < 1

in the critical strip to another such case, reflected in the line σ = ½. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole [[compl