The paper is devoted to Epstein zeta-function (Formula presented.), (Formula presented.), of a quadratic form of three variables. We obtain a probabilistic characterization of the function (Formula presented.) in terms of weak convergence of probability measures in the space of analytic functions defined by means of shifts (Formula presented.), where (Formula presented.) is a differentiable function increasing to infinity with a monotonic derivative satisfying certain growth conditions. The Gram function is an example of (Formula presented.). We prove a limit theorem for (Formula presented.) with an explicitly given limit measure—the distribution of an analytic random element. For the proof, the mean square estimate for (Formula presented.) is applied. The obtained limit theorem characterizes the chaotic behavior of the function (Formula presented.) in the space of analytic functions and is applied for approximation of a certain class of analytic functions by shifts (Formula presented.). This shows the impact of (Formula presented.) in approximation function theory.