Joint value-distribution of shifts of the Riemann zeta-function

We prove that any non-zero complex values $z_1,\ldots,z_n$ can be approximated by the following integral shifts of the Riemann zeta-function $\zeta(s+id_1\tau),\ldots,\zeta(s+id_n\tau)$ for infinitely many $\tau$, provided $d_1,\ldots,d_n\in\mathbb{N}$ and $s$ is a fixed complex number lying in the right open half of the critical strip.