The generalized Reimann hypothesis (GRH) is a stronger form of the Reimann hypothesis that extends it to all L-functions instead of just the Reimann zeta function. The precise statement is as follows:

Let χ be a function from the positive integers to the complex numbers meeting the following criteria:

• χ(1) = 1, and χ(ab) = χ(a)χ(b) for all a, b.

• χ is periodic, and χ(n) = 0 for any n that isn't coprime with the period.

Then a function L(χ,s) can be defined as the sum over all positive integers n of χ(n)/n^s, wherever this sum is defined, and extended to a meromorphic function on the complex plane by analytic continuation.

The GRH is the claim that, for every χ and s such that L(χ,s) = 0, either s is a negative real number, or or the real part of s is 1/2.