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 (10⁸q, A 10⁷q+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.