Distribution of lattice points on hyperbolic surfaces

Let two lattices ${\displaystyle \Lambda \prime ,\Lambda \prime \prime \subset {ℝ}^s}$ have the same number of points on each hyperbolic surface ${\displaystyle |x₁...x_s|=C}$. We investigate the case when Λ', Λ'' are sublattices of ${\displaystyle {\mathbb{Z}}^s}$ of the same prime index and show that then Λ' and Λ'' must coincide up to renumbering the coordinate axes and changing their directions.