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Zeros of the Riemann zeta function and its derivatives have been studied bymany mathematicians. Among, the number of zeros and the distribution of thereal part of non-real zeros of the derivatives of the Riemann zeta functionhave been investigated by Berndt, Levinson, Montgomery, Akatsuka, and theauthor. Berndt, Levinson, and Montgomery investigated the unconditional case,while Akatsuka and the author gave sharper estimates under the truth of theRiemann hypothesis. In this paper, we prove similar results related to thefirst derivative of the Dirichlet $L$-functions associated with primitiveDirichlet characters under the assumption of the generalized Riemannhypothesis.

Two estimates on the distribution of zeros of the first derivative of Dirichlet $L$-functions under the generalized Riemann hypothesis