I am trying to find a function, that assuming RH, generates subsequent non-trivial zeros $\rho_n$ in an alternating way i.e.:

$$\frac12+14.134...i,\frac12-21.022...i,\frac12+25.010...i, \dots$$

or it generates the conjugated equivalent of this sequence.

I expect that such a function would still require $\zeta(s)$ as a component, since it currently is the only known function that induces $\rho_n$'s. However, it now also requires further distinguishing information for each 'subsequent' zero about whether it should be generated or be 'suppressed'.

That extra bit of information does not appear to be derivable from $\zeta(s)$ itself, but I do believe that the Riemann $\xi(s) = \frac12 s(s-1) \pi^{-\frac{s}{2}} \Gamma(\frac{s}{2}) \zeta(s)$, does contain it. This becomes clearer when using $\Xi(t)=\xi(s)$ where $s=\frac12 + ti$ with $s \in \mathbb{C}, t \in \mathbb{R}$.

In that case:

$$\displaystyle \zeta\left(1 \pm \dfrac{|\Xi'(t)|}{2\Xi'(t)}+ti \right)$$

does more or less what I need, however it is a pretty ugly construct.

Any ideas/hints/tips on how to find a more elegant function?


P.S.: Background for my question here


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