Let $K=\mathbb{Q}\left(\sqrt{-D}\right)$ be an imaginary quadratic field, and denote by $h$ its class number. It is shown that there is an absolute constant $c\>0$ such that for sufficiently large $D$ at least $c·h{\prod }_{p\mid D}(1-p^{-1})$ of the $h$ distinct $L$-functions $L_K(s,\chi )$ do not vanish at the central point $s=1/2$.