We work on a GRB sample whose initial Lorentz factors (Γ 0 ) are constrained with the afterglow onset method and the jet opening angles (θ j ) are determined by the jet break time. We confirm the Γ 0 -E γ,iso correlation by Liang et al. (2010), and the Γ 0 -L γ,iso correlation by Lü et al. (2012). Furthermore, we find correlations between Γ 0 and the beaming corrected γ-ray energy (E γ ) and mean γ-ray luminosity (L γ ). By also including the kinetic energy of the afterglow, we find rough correlations (with larger scatter) between Γ 0 and the total (γ-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 νν -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.