  The Bishop invariant is a powerful tool in the analysis of real submanifolds
of complex space that associates to every (non-degenerate) complex tangent of
the embedding a non-negative real number (or infinity). It is a biholomorphism
invariant that gives information regarding the local hull of holomorphy of the
manifold near the complex tangent. In this paper, we derive a readily
applicable formula for the computation of the Bishop invariant for graphical
embeddings of 3-manifolds into $\mathbb{C}^3$. We then exhibit some examples
over $S^3$ demonstrating the different possible configurations of the Bishop
invariant along complex tangents to such embeddings. We will also generate a
few more results regarding the behavior of the Bishop invariant in certain
situations. We end our paper by analyzing the different possible outcomes from
the perturbation of a degenerate complex tangent.
