Explicit bounds on exceptional zeroes of Dirichlet L-function II

This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L^{\mathrm{\prime }}(\sigma ;\chi )$ for $\sigma$ close to unity, using a result on the average of Dirichlet characters, and on the lower bound for $L(1;\chi )$, with computational aid.