For all sufficiently large primes q, the lowest non-trivial zero of Dirichlet L-functions modulo q has height below 1075 times the average spacing, with a positive proportion having even smaller first zeros.
Explicit zero-free regions for the Riemann zeta-function
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Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros
For all sufficiently large primes q, the lowest non-trivial zero of Dirichlet L-functions modulo q has height below 1075 times the average spacing, with a positive proportion having even smaller first zeros.