Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of:
$$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$
are on the critical line $\Re(s)=\frac12$ for all $z \lt 1$ (with both individual terms $\ne 0$, except when $z=-1$).
I derived two integral expressions for this function that both only converge in the strip for $z \lt 1$,
$$\displaystyle \int_0^\infty \frac{t^ {s-1} \, \pm \, t^{-s}}{\mathrm{e}^{t}-z}\mathrm{d}t$$
and
$$\displaystyle \int_0^1 \frac{\ln^{s-1}\left(\frac{1}{t}\right) \, \pm \, \ln^{-s}\left(\frac{1}{t}\right)} {1-z\,t}\mathrm{d}t$$
however neither does reveal any further information about their zeros.
Notes:
when $z \rightarrow 0$, the function reduces to $\Gamma(s) \pm \Gamma(1-s)$, for which it has been proven here that all complex zeros in the strip are on the critical line (and only a finite number lie outside the strip). Additional observation: I found that by gradually declining $z$ from $-0$ to $-1$ and for each step calculating the 'adjacent' zeros (that I still find to all reside on the line $\Re(s)=\frac12$), their imaginary parts seem to follow a unique path connecting a zero of $\Gamma(s) \pm \Gamma(1-s)$ to a zero of the function at $z=-1$ (i.e. including the non-trivial Riemann zeros). Tested only the first few paths (used $\pm=-$ and decreased $z$ in steps of $0.1$), but the emerging pattern looks strong (would love to produce a, now 2-D, version of this one).
unlike for $\Gamma(s)$ and $\zeta(s)$, there doesn't seem to exist a reflection formula relating $Li_s(z)$ and $Li_{1-s}(z)$.
Questions:
Are there known counter examples of complex zeros lying off the critical line, but still in the strip (I know there do exist a few complex zeros outside the strip and also that there are real zeros)?
Is this phenomenon a consequence of the RH or would a prove of it imply the RH (note that $z=-1$ turns the $Li_s(z)$ into the Dirichlet $\eta(s)$-function that can be directly related to $\zeta(s)$)?
Thanks.