  A finite length graded $R$-module $M$ has the Weak Lefschetz Property if
there is a linear element $\ell$ in $R$ such that the multiplication map
$\times\ell: M_i\to M_{i+1}$ has maximal rank. The set of linear forms with
this property form a Zariski-open set and its complement is called the
non-Lefschetz locus. In this paper we focus on the study of the non-Lefschetz
locus for the first cohomology module $H_*^1(\mathbb{P}^2,\mathcal{E})$ of a
locally free sheaf $\mathcal{E}$ of rank $2$ over $\mathbb{P}^2$. The main
result is to show that this non-Lefschetz locus has the expected codimension
under the assumption that $\mathcal{E}$ is general.
