  The degenerate leptogenesis is studied when the degeneracy in two of the
heavy right-handed neutrinos [the third one is irrelevant if $\mu -\tau $
symmetry is assumed] is due to $\bar{L}\equiv (L_{e}-L_{\mu}-L_{\tau}) $
discrete symmetry. It is shown that a sizeable leptogenesis asymmetry
$(\epsilon \succsim 10^{-6}) $ is possible. The level of degeneracy required
also predicts the Majorana phase needed for the asymmetry. Since it is the same
phase, which appears in the double $\beta $% -decay and this prediction is
testable. Implication of non-zero reactor angle $\theta_{13}$ are discussed. It
is shown that the contribution from $% \sin ^{2}\theta_{13}$ to leptogenesis
asymmetry parameter may even dominate.An accurate measument of $\sin
^{2}\theta_{13}$ would have important implications for the mass degeneracy of
heavy right-handed neutrinos.
