Low dimensional cohomology of general conformal algebras $g{c}_N$

We compute the low dimensional cohomologies ${\tilde{H}}^q(g{c}_N,C)$, $H^q(g{c}_N,\text{C})$ of the infinite rank general Lie conformal algebras $g{c}_N$ with trivial coefficients for $q\le 3,N=1$ or $q\le 2,N\ge 2$. We also prove that the cohomology of $g{c}_N$ with coefficients in its natural module is trivial, i.e., $H^{\ast }(g{c}_N,\text{C}[\text{ptl}{]}^N)=0$; thus partially solve an open problem of Bakalov-Kac-Voronov in [{\it Comm. Math. Phys.,} {\bf200} (1999), 561-598].