  We select a sample of 34 gamma-ray bursts (GRBs) whose $\Gamma_0$ values are
derived with the onset peaks observed in the afterglow lightcurves (except for
GRB 060218 whose $\Gamma_0$ is estimated with its radio data), and investigate
the correlations among $\Gamma_0$, the isotropic peak luminosity ($L_{\rm
iso}$), and the peak energy ($E_{\rm p,z}$) of the $\nu f_\nu$ spectrum in the
cosmological rest frame. An analysis of pair correlations among these
observables well confirms the results reported by the previous papers. More
interestingly, a tight correlation among $L_{\rm iso}$, $E_{\rm p,z}$, and
$\Gamma_0$ is found from a multiple regression analysis, which takes the form
of $L_{\rm iso} \propto E_{\rm p,z}^{1.34\pm 0.14} \Gamma_0^{1.32\pm 0.19}$ or
$E_{\rm p,z} \propto L_{\rm iso}^{0.55\pm 0.06}\Gamma_0^{-0.50\pm 0.17}$. Nine
other GRBs whose $\Gamma_0$ are derived via the pair production opacity
constraint also follow such a correlation. Excluding GRB 060218, the $L_{\rm
iso}-E_{\rm p,z}-\Gamma_0$ correlation is valid, and it even holds in the jet
co-moving frame. However, GRB 060218 deviates the $L^{'}_{\rm iso}-E^{'}_{\rm
p}$ relation of typical GRBs in the jet co-moving frame with $3\sigma$. We
argue that the $L_{\rm iso} - E_{\rm p, z} - \Gamma_0$ correlation may be more
physical than the $L_{\rm iso} - E_{\rm p,z}$ correlation, since physically the
relationship between the observed $L_{\rm iso}$ and $E_{\rm p,z}$ not only
depends on radiation physics, but also depends on the bulk motion of the jet.
We explore the possible origins of this correlation and discuss its physical
implications for understanding GRB jet composition and radiation mechanism.
