In the paper, we prove the discrete universality theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the L-functions of elliptic curves. We consider an approximation of analytic functions by translations LE (s+imh) , where h > 0 is a fixed number, the translations of the imaginary part of the complex variable take values from some discrete set such as arithmetical progression. We suppose that the number h > 0 is chosen so that exp{2πk/h } is an rational number for some non-zero integer. The proof of discrete universality of the derivatives of L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.