Long-duration gamma-ray bursts (GRBs) are thought to be connected with
luminous and energetic supernovae [SNe, called hypernovae (HNe)] resulting
from the black-hole (BH) forming collapse of massive stars. For recent
nearby GRBs~060505 and 060614, however, the expected SNe have not been
detected. The upper limits to the SN brightness are about 100 times fainter
than GRB-associated HNe (GRB-HNe), corresponding to the upper limits to the
ejected 56Ni masses of M(56Ni)~10^{-3} Msun. SNe with a small amount of
56Ni ejection are observed as faint Type II SNe. HNe and faint SNe are
thought to be responsible for the formation of extremely metal-poor
(EMP) stars. In this paper, a relativistic jet-induced BH forming explosion
of the 40 Msun star is investigated and hydrodynamic and nucleosynthetic
models are presented. These models can explain both GRB-HNe and GRBs
without bright SNe in a unified manner. Their connection to EMP stars is
also discussed. We suggest that GRBs without bright SNe are likely to
synthesize M(56Ni)~10^{-4}-10^{-3} Msun or ~10^{-6} Msun.