  Let $K$ be an algebraically closed field of characteristic zero and let $R =
K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$. Let $A_n(K)$ be the $n^{th}$
Weyl algebra over $K$. By a result of Lyubeznik, the local cohomology modules
$H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this paper we
compute the Euler characteristic of De-Rahm cohomology of $H^{\htt P}_P(R)$ for
certain classes of prime ideals $P$ in $R$.
