  Classical Cepheids (DCEPs) play a fundamental role in the calibration of the
extra-galactic distance ladder which eventually leads to the determination of
the Hubble constant($H_0$) thanks to the period-luminosity ($PL$) and
period-Wesenheit ($PW$) relations exhibited by these pulsating variables.
Therefore, it is of great importance to establish the dependence of $PL/PW$
relations on metallicity. We aim at quantifying the metallicity dependence of
the Galactic DCEPs' $PL/PW$ relations for a variety of photometric bands
ranging from optical to near-infrared. We gathered a literature sample of 910
DCEPs with available [Fe/H] values from high-resolution spectroscopy or
metallicities from \gaia\ Radial Velocity Spectrometer. For all these stars, we
collected photometry in the $G_{BP},G_{RP},G,I,V,J,H,K_S$ bands and astrometry
from the \gaia\ DR3. These data have been used to investigate the metal
dependence of both intercepts and slopes of a variety of $PL/PW$ relations at
multiple wavelengths. We find a large negative metallicity effect on the
intercept ($\gamma$ coefficient) of all the $PL/PW$ relations investigated in
this work, while present data still do not allow us to draw firm conclusions
regarding the metal dependence of the slope ($\delta$ coefficient). The typical
values of $\gamma$ are around $-0.4:-0.5$ mag/dex, i.e. larger than most of the
recent determinations present in the literature. We carried out several tests
which confirm the robustness of our results. As in our previous works, we find
that the inclusion of global zero point offset of \gaia\ parallaxes provides
smaller values of $\gamma$ (in an absolute sense). However, the assumption of
the geometric distance of the LMC seems to indicate that larger values of
$\gamma$ (in an absolute sense) would be preferred.
