Regarding Jonah's remark about plotting the inverse images of n. A quick way to visualize the zeros of a meromorphic function $f(s)$ is to plot (by color-coding with four colors) the quadrant of the value of $f(s)$. Points at the junction of four differently colored regions are either zero or poles. For $f(s) = \zeta(s) - n$ there is only one pole, so all the other 4-color junctions are inverse images of $n$. The plots I've looked at show some behavior that looks related to many other plots one has seen in connection with zeta. But the plot of the inverse image of the set of all (Gaussian) integers obtained from $f(s) = Mod[\zeta(s),1]$ (using $Mathematica$ notation) seems particularly interesting, especially (for example) in the three by three square with center 1, or in much smaller regions in the left half plane, for example, the square centered at $-25 + \frac 12 i$ with side length $10^{-5}$.
Regarding Jonah's remark about plotting the inverse images of n. A quick way to visualize the zeros of a meromorphic function $f(s)$ is to plot (by color-coding with four colors) the quadrant of the value of $f(s)$. Points at the junction of four differently colored regions are either zero or poles. For $f(s) = \zeta(s) - n$ there is only one pole, so all the other 4-color junctions are inverse images of $n$. The plots I've looked at show some behavior that looks related to many other plots one has seen in connection with zeta. But the plot of the inverse image of the set of all integers obtained from $f(s) = Mod[\zeta(s),1]$ (using $Mathematica$ notation) seems particularly interesting, especially (for example) in the three by three square with center 1, or in much smaller regions in the left half plane, for example, the square centered at $-25 + \frac 12 i$ with side length $10^{-5}$.