In this paper we show that, under the assumption that all the zeros of the L-functions under consideration are either real or lie on the critical line, one may considerably improve on the known results on Landau-Siegel zeros.
"Some remarks on Landau-Siegel zeros." Duke Math. J. 111 (3) 495 - 507, 15 February 2002. https://doi.org/10.1215/S0012-7094-02-11134-X