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We select a sample of 34 gamma-ray bursts (GRBs) whose $\Gamma_0$ values arederived with the onset peaks observed in the afterglow lightcurves (except forGRB 060218 whose $\Gamma_0$ is estimated with its radio data), and investigatethe correlations among $\Gamma_0$, the isotropic peak luminosity ($L_{\rmiso}$), and the peak energy ($E_{\rm p,z}$) of the $\nu f_\nu$ spectrum in thecosmological rest frame. An analysis of pair correlations among theseobservables well confirms the results reported by the previous papers. Moreinterestingly, a tight correlation among $L_{\rm iso}$, $E_{\rm p,z}$, and$\Gamma_0$ is found from a multiple regression analysis, which takes the formof $L_{\rm iso} \propto E_{\rm p,z}^{1.34\pm 0.14} \Gamma_0^{1.32\pm 0.19}$ or$E_{\rm p,z} \propto L_{\rm iso}^{0.55\pm 0.06}\Gamma_0^{-0.50\pm 0.17}$. Nineother GRBs whose $\Gamma_0$ are derived via the pair production opacityconstraint also follow such a correlation. Excluding GRB 060218, the $L_{\rmiso}-E_{\rm p,z}-\Gamma_0$ correlation is valid, and it even holds in the jetco-moving frame. However, GRB 060218 deviates the $L^{'}_{\rm iso}-E^{'}_{\rmp}$ relation of typical GRBs in the jet co-moving frame with $3\sigma$. Weargue that the $L_{\rm iso} - E_{\rm p, z} - \Gamma_0$ correlation may be morephysical than the $L_{\rm iso} - E_{\rm p,z}$ correlation, since physically therelationship between the observed $L_{\rm iso}$ and $E_{\rm p,z}$ not onlydepends on radiation physics, but also depends on the bulk motion of the jet.We explore the possible origins of this correlation and discuss its physicalimplications for understanding GRB jet composition and radiation mechanism.

A Tight $L_{\rm p, iso}-E'_{\rm p}-\Gamma_0$ Correlation of Gamma-Ray Bursts