  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 the
relation 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.
