About one functional equation

The hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations is studied. A functional equation is found for the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations in the case of rational $\beta$, which sets an analytical continuation on the entire complex plane, except for the point $\alpha=1$, in which the pole is of the first order.
The found functional equation allows us to raise the question of continuity for the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations in the case of rational $\beta$.