  It is proposed that the luminosity function, the rest-frame spectral
correlations and distributions of cosmological Long-duration (Type-II)
Gamma-Ray Bursts (LGRBs) may be very well described as multivariate log-normal
distribution. This result is based on careful selection, analysis and modeling
of LGRBs' temporal and spectral variables in the largest catalog of Gamma-Ray
Bursts available to date: 2130 BATSE GRBs, while taking into account the
detection threshold and possible selection effects. Constraints on the joint
rest-frame distribution of the isotropic peak luminosity (Liso), total
isotropic emission (Eiso), the time-integrated spectral peak energy (Epkz) and
duration (T90z) of LGRBs are derived. The presented analysis provides evidence
for a relatively large fraction of LGRBs that have been missed by BATSE
detector with Eiso extending down to ~ 10^49 [erg] and observed spectral peak
energies (Epk) as low as ~ 5 [keV]. LGRBs with rest-frame duration T90z < 1 [s]
or observer-frame duration T90 < 2 [s] appear to be rare events (<0.1% chance
of occurrence). The model predicts a fairly strong but highly significant
correlation (rho=0.58 \pm 0.04) between Eiso & Epkz of LGRBs. Also predicted
are strong correlations of Liso & Eiso with T90z and moderate correlation
between Liso & Epkz. The strength and significance of the correlations found,
encourage the search for underlying mechanisms, though undermine their
capabilities as probes of Dark Energy's equation of state at high redshifts.
The presented analysis favors -- but does not necessitate -- a cosmic rate for
BATSE LGRBs tracing metallicity evolution consistent with a cutoff ~ 0.2-0.5,
assuming no luminosity-redshift evolution.
