Is there an analytic function with zeroes only at:
- every $-2n$,
- $\frac12\pm it$, and
- at least one at $\frac12\pm\epsilon\pm it$ where $0<\epsilon<\frac12, t\neq0$ (and these zeroes observing the known reflective symmetries within the critical strip)?
A answer assuming the Riemann hypothesis true would be fine.
To my mind the uniqueness of any analytic continuation of $\zeta(s)$ is suggestive of the existence of such a function being incompatible with the Riemann Hypothesis.
If not, uniqueness of $\zeta$ is of course a nice simple sufficiency for the Riemann hypothesis.