Special values of $\zeta$ outside the real line and the critical strip The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (and outside of critical strip)?
Conjectured irrationality, mysteriously neat expressions or something of the like.
Edit. Since the question is necessarily broad, I'll try to make it a bit more clear:
Is there any $s\in \mathbb{C}$, $\mathfrak{I}(s)\neq0$, $\mathfrak{R}(s)>1$ such that $s$ or its real/imaginary components are suspected to be transcendental, or to be give any new arithmetical information about $\mathbb{Q}$?
 A: There is a well known conjecture that the ordinates of the non-trivial zeros of $\zeta$ are $\mathbb{Q}$-linearly independent. 
There are two major motivations for this conjecture. First all numbers which are not obviously rational are irrational, second linear independence together with RH implies that the suitably renormalized error term in the prime number theorem behaves like a sum of independent random variables, so we can easily compute its distribution.
A: This answer is more philosophical than mathematical, but I decided to take a risk today...
One could also conjecture that actually all interesting information is only contained in the non-trivial zeros in the critical strip. The logic would be that on top of encoding the distribution of primes, the non-trivial zeros $\rho$ in the Hadamard product: 
$$\displaystyle \zeta(s)=\frac{\pi^{\frac{s}{2}}\,\prod_{\rho}\left(1-\frac{s}{\rho}\right)}{2\,(s-1)\,\Gamma(1+\frac{s}{2})}$$
also encode (with some help) all the "trivial" results at the integers. 
The distinction between "trivial" and "non-trivial" zeros, is a label driven by our human perception of 'what we can simply explain' and 'what we can't easily understand'. So, there could be many interesting "abstract" results outside the strip, that don't yet resonate in any way with our current state of (even the abstract) mathematical knowledge. Poetically labelling this domain as "a suspicious waste of space" would hopefully one day turn into "you'd ain't seen nothing yet".
The more I understand about analytic continuation, the more I also start to wonder whether a more appropriate term would actually be: 'meaningful analytic discontinuation'. My logic is that an analytically continued function like $\zeta$ is valid in a much wider and richer domain, than the limited domain of convergence (e.g. $\Re(s)>1$) that we typically start from and frame up in terms of infinite sums, integrals and/or (Euler) products based on powers of integers/primes. 
Having worked quite a bit with the Zeta-function, I actually start to truly appreciate the beauty (or better "L'affreuse beauté" as the French would say) of the non-trivial zeros and them possible all lying on the critical line. Yes, I start to like them even more than the primes themselves (I know, that is a very subjective statement). This feeling is reinforced by the Hadamard product above, that also allows the Euler product of primes to be encoded by the $\rho's$ (no chance you could do the same with the primes, although of course they deserve credit for encoding all the integers :-) ). 
It might be similar to the journey physics went through (I know, they invented Zeta regularization. A method that is not considered correct in pure mathematics), where many scientists still try to align the counter intuitive aspects of Quantum Mechanics with the 'classical limit' of the 3D world we live in. 
Speaking of physics, we could also slightly rephrase your question as: "Whether any special values of $\zeta$ exist outside the real line and the critical strip, show up in physics?" Again the answer seems no, since $\zeta(-1)=-\frac{1}{12}$ (again an integer) shows up in the Casimir force (although an answer to this question suggests that $\zeta'(-\frac12)$ also plays a role in this effect) and potentially the distribution of the imaginary parts of the $\rho's$ (the strip again) being correlated to eigenvalues of random GUE matrices used in Quantum Mechanics. I don't know of any other examples.
