The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - one can give some analytic study trying to give an effective result. But nowadays we can also try to directly evaluate and verify it with a computer - the thing is that it turns out to be really hard to do that.

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important point out and compare, for instance, the following graph of $log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in?

(Note also direct implication to zeros of zeta - RH, for instance.)