'Almost' universality of the Lerch zeta-function

The Lerch zeta-function $L(lambda,alpha,s)$ with transcendental parameter $alpha$, or with rational parameters $alpha$ and $lambda$ is universal, i.e., a wide class of analytic functions is approximated by shifts $L(lambda,alpha,s+itau)$, $tau in mathbb{R}$. The case of algebraic irrational $alpha$ is an open problem. In the paper, it is proved that, for all parameters $alpha$, $0