Dirichlet L-function - Zeros of The Dirichlet L-functions

Zeros of The Dirichlet L-functions

If χ is a primitive character with χ(−1) = 1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative even integers. If χ is a primitive character with χ(−1) = −1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative odd integers.

Up to the possible existence of a Siegel zero, zero-free regions including and beyond the line Re(s) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions.

Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis, so the Dirichlet L-functions are conjectured to obey the generalized Riemann hypothesis.