The function:
$$f(s) ={\frac { \left( \zeta \left( s \right) \right) ^{ \left( \zeta \left( 1-s \right) \right) ^{-1}}}{ \left( \zeta \left( 1-s \right) \right) ^{ \left( \zeta \left( s \right) \right) ^{-1}}}}$$
gives the following plot for $\Re\left(f(\frac12+i\,t) \right)$:
The function heavily oscillates around each non-trivial zero ($\rho$). I wondered about the 'width' of these oscillations and from plots a bit higher up, e.g.:
it appears that the nearer the $\rho$s are to each other, the wider spread the oscillations around them are. The $\rho$s in this domain:
141.123707404 143.111845808 146.000982487 147.422765343 150.053520421 150.925257612 153.024693811 156.112909294 157.597591818 158.849988171
I know that $\rho$s tend to repel each other, so could there be a correlation between the distance to their neighbour(s) and the 'width' (or 'energy') of the oscillations that $f(s)$ induces around them?
ADDED:
Tested many more zeros and the same pattern emerges. It is also visible at the first Lehmer pair $\gamma_{6709}, \gamma_{6710}$ i.e. $t=7005.06..., 7005.10...$. Between these closely adjacent zeros there still is a clear demarcation between the (fierce) oscillations around each zero (right graph). The red arrows indicate that when two zeros reside very close together, the "peak in the middle" is no longer 1 but shrinks to a very small "hill". Makes me wonder whether by establishing the "fierceness" of the oscillations around a $\rho_n$, we could roughly predict its distance to its neighbouring $\rho_{n+1}$? (note: graphs only show $\left(\Re(f(s)\right)$ ).
