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\#P-hardness of computing matrix immanants are proved for each member of abroad class of shapes and restricted sets of matrices. We prove \#P-hardness ofcomputing $\lambda$-immanants of $0$-$1$ matrices when $\lambda$ has a largedomino-tilable part and satisfying some technical conditions. We also givehardness proofs of some $\lambda$-immanants of weighted adjacency matrices ofplanarly drawable directed graphs, such that the shape $\lambda =(\mathbf{1}+\lambda_d)$ has size $n$ such that $|\lambda_d| = n^{\varepsilon}$for some $0<\varepsilon<\frac{1}{2}$, and for some $w$, the shape$\lambda_d/(w)$ is tilable with $1\times 2$ dominos.

#P-hardness proofs of matrix immanants evaluated on restricted matrices