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Let $f$ be a holomorphic cusp form of weight $k$ with respect to$SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the$L$-function attached to the form $f$. In this paper, we shall give therelation of the number of zeros of $L_f(s)$ and the derivatives of $L_f(s)$using Berndt's method, and an estimate of zero-density of the derivatives of$L_f(s)$ based on Littlewood's method.

Distribution of zeros and zero-density estimates for the derivatives of $L$-functions attached to cusp forms