  We work on a GRB sample whose initial Lorentz factors ($\Gamma_0$) are
constrained with the afterglow onset method and the jet opening angles
($\theta_{\rm j}$) are determined by the jet break time. We confirm the
$\Gamma_0$ - $E_{\gamma,\rm iso}$ correlation by Liang et al. (2010), and the
$\Gamma_0$ - $L_{\gamma,\rm iso}$ correlation by L{\"u} et al. (2012).
Furthermore, we find correlations between $\Gamma_0$ and the beaming corrected
$\gamma$-ray energy ($E_{\gamma}$) and mean $\gamma$-ray luminosity
($L_{\gamma}$). By also including the kinetic energy of the afterglow, we find
rough correlations (with larger scatter) between $\Gamma_0$ and the total
($\gamma$-ray plus kinetic) energy and the total mean luminosity, both for
isotropic values and beaming corrected values: these correlations allow us to
test the data with GRB central engine models. Limiting our sample to the GRBs
that likely have a black hole central engine, we compare the data with
theoretical predictions of two types of jet launching mechanisms from BHs, i.e.
the non-magnetized $\nu \bar\nu$ - annihilation mechanism, and the strongly
magnetized Blandford-Znajek (BZ) mechanism. We find that the data are more
consistent with the latter mechanism, and discuss the implications of our
findings for GRB jet composition.
