Summary: In this paper, we study the number of additional zeros of Dirichlet \(L\)-function caused by multiplicity by using asymptotic large sieve. Then in asymptotic terms we prove that there are \(> 80.13\%\) of zeros of the family of Dirichlet \(L\)-functions which are distinct and \(> 60.261\%\) of zeros of the family of Dirichlet \(L\)-functions which are simple. In addition, assuming the generalized Riemann hypothesis, we improve these proportions to 83.216% and 66.433%.