  We study the use of the complex-Langevin equation (CLE) to simulate lattice
QCD at a finite chemical potential ($\mu$) for quark-number, which has a
complex fermion determinant that prevents the use of standard simulation
methods based on importance sampling. Recent enhancements to the CLE specific
to lattice QCD inhibit runaway solutions which had foiled earlier attempts to
use it for such simulations. However, it is not guaranteed to produce correct
results. Our goal is to determine under what conditions the CLE yields correct
values for the observables of interest. Zero temperature simulations indicate
that for moderate couplings, good agreement with expected results is obtained
for small $\mu$ and for $\mu$ large enough to reach saturation, and that this
agreement improves as we go to weaker coupling. For intermediate $\mu$ values
these simulations do not produce the correct physics. We compare our results
with those of the phase-quenched approximation. Since there are indications
that correct results might be obtained if the CLE trajectories remain close to
the $SU(3)$ manifold, we study how the distance from this manifold depends on
the quark mass and on the coupling. We find that this distance decreases with
decreasing quark mass and as the coupling decreases, i.e. as the simulations
approach the continuum limit.
