Discrete universality of the $L$-functions of elliptic curves

A discrete universality theorem is obtained in the Voronin sense for the $L$-functions of elliptic curves. We use the difference of an arithmetical progression $h>0$ such that $\exp\{\frac{2\pi k}h\}$ is rational for some $k\ne0$. A limit theorem in the space of analytic functions plays a crucial role in the proof.