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\leq 400\,000
. We check, to height,
max
(
10
8
q
,
A
⋅
10
7
q
+
200
)
\textrm {max}\left (\frac {10^8}{q},\frac {A\cdot 10^7}{q}+200\right )
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\leq 2\,000\,000
vanishes at its central point.