  We compared redshifts $z_Y$ from Yonetoku relation and $z_{lag}$ from the
lag-luminosity relation for 565 BASTE GRBs and were surprised to find that the
correlation is very low. Assuming that the luminosity is a function of both
$z_Y$ and the intrinsic spectral lag $\tau_{lag}$, we found a new redshift
dependent lag-luminosity relation as $L=7.5\times 10^{50}{\rm
erg/s}(1+z)^{2.53}\tau_{lag}^{-0.282}$ with the correlation coefficient of 0.77
and the chance probability of $7.9\times 10^{-75}$. To check the validity of
this method, we examined the other luminosity indicator, Amati relation, using
$z_Y$ and the observed fluence and found the correlation coefficient of 0.92
and the chance probability of $5.2\times 10^{-106}$. Although the spectral lag
is computed from two channels of BATSE, our new lag-luminosity relation
suggests that a possible lag-luminosity relation in the \swift era should also
depend on redshift.
