  We consider two empirical relations using data only from the prompt emission
of Gamma-Ray Bursts (GRBs), peak energy ($E_p$) - peak luminosity ($L_p$)
relation (so called Yonetoku relation) and $E_p$-isotropic energy ($E_{\rm
iso}$) relation (so called Amati relation). We first suggest the independence
of the two relations although they have been considered similar and dependent.
  From this viewpoint, we compare constraints on cosmological parameters,
$\Omega_m$ and $\Omega_{\Lambda}$, from the Yonetoku and Amati relations
calibrated by low-redshift GRBs with $z < 1.8$. We found that they are
different in 1-$\sigma$ level, although they are still consistent in 2-$\sigma$
level. This and the fact that both Amati and Yonetoku relations have systematic
errors larger than statistical errors suggest the existence of a hidden
parameter of GRBs. We introduce the luminosity time $T_L$ defined by $T_L\equiv
E_{\rm iso}/L_p$ as a hidden parameter to obtain a generalized Yonetoku
relation as $(L_p/{10^{52} \rm{erg s^{-1}}}) =
  10^{-3.88\pm0.09}(E_p/{\rm{keV}})^{1.84\pm0.04}
  (T_L/{\rm{s}})^{-0.34\pm0.04}$. The new relation has much smaller systematic
error, 30%, and can be regarded as "Fundamental plane" of GRBs. We show a
possible radiation model for this new relation. Finally we apply the new
relation for high-redshift GRBs with $1.8 < z < 5.6$ to obtain
$(\Omega_m,\Omega_{\Lambda}) =
  (0.16^{+0.04}_{-0.06},1.20^{+0.03}_{-0.09})$, which is consistent with the
concordance cosmological model within 2-$\sigma$ level.
