  In this paper it is explained how one can construct non-selfdual
4-dimensional $\ell$-adic Galois representations of Hodge type
$h^{3,0}=h^{2,1}=h^{1,2}=h^{0,3}=1$, assuming a hypothesis concerning the
cohomology of a certain threefold. For one such a representation the first
80000 coefficients of its $L$-function are computed, and it is numerically
verified that this $L$-function satisfies a functional equation. Also a
candidate for the conductor is obtained.
