Arguing security of an encryption scheme
If the adversary knows $L$, then he can find $L^{-1}$, the multiplicative inverse of $L$ modulo $q$ where $q$ is the order of $g$. Then he can compute $g^r =((g^L)^r)^{L^{-1}}$. Once he find $g^r$, he can compute $m$ by eliminating $g^r$ from $m\cdot g^r$.

If the adversary knows only $g^L$, then it is ok. This case is like El Gamal. Let $h=g^L$, what you reveal is $(h^r,m\cdot g^r)$. Your private key is $L^{-1}$.