  In this talk we report on a recent lattice investigation of the Landau gauge
gluon and ghost propagators in pure SU(3) lattice gauge theory with a special
emphasis on the Gribov copy problem. In the (infrared) region of momenta $q^2
\le 0.3 \mathrm{GeV}^2$ we find the corresponding MOM scheme running coupling
$\alpha_s(q^2)$ to rise in $q$. We also report on a first SU(3) computation of
the ghost-gluon vertex function showing that it deviates only weakly from being
constant. In addition we study the spectrum of low-lying eigenvalues and
eigenfunctions of the Faddeev-Popov operator as well as the spectral
representation of the ghost propagator.
