  The existence of self-intersection local time (SILT), when the time diagonal
is intersected, of the $(\alpha,d,\beta)$-superprocess is proved for
$d/2<\alpha $ and for a renormalized SILT when $d/(2+(1+\beta)^{-1})<\alpha
\leq d/2$. We also establish Tanaka-like formula for SILT.
