Zeros of the Derivatives of $L$-functions Attached to Cusp Forms

Let $f$ be a holomorphic cusp form of weight $k$ with respect to $\mathrm {SL}_2(\mathbb {Z})$ which is a normalized Hecke eigenform, and $L_f(s)$ the $L$-function attached to $f$. We shall give a relation between the number of zeros of $L_f(s)$ and of 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.