  Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$.
Suppose there exists be a non-zero $A$ module $M$ of finite length and finite
projective dimension such that $\ell\ell(M)$, the Lowey length of $M$, is equal
to $\lambda(M)$, the length of $M$. Then we show that necessarily $A$ is at
worst a hypersurface singularity. We also characterize Gorenstein local rings
having a non-zero module $M$ of finite length and finite projective dimension
with $\ell\ell(M) = \lambda(M)-1$.
