Numerical computations concerning the GRH

We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q ≤ 400 000 q≤ 400\,000 . We check, to height, max ( 10 8 q , A ⋅ 10 7 q + 200 ) max (10^8/q,A· 10^7/q+200 ) with A = 7.5 A=7.5 in the case of even characters and A = 3.75 A=3.75 for odd characters. In addition we confirm that no Dirichlet L-function with a modulus q ≤ 2 000 000 q≤ 2\,000\,000 vanishes at its central point.