  We compute the static potential in the Gribov-Zwanziger Lagrangian as a
function of the Gribov mass, gamma, in the MSbar scheme in the Landau gauge at
one loop. The usual gauge independent one loop perturbative static potential is
recovered in the limit as gamma -> 0. By contrast the Gribov-Zwanziger static
potential contains the term gamma^2/(p^2)^2. However, the linearly rising
potential in coordinate space as a function of the radial variable r does not
emerge due to a compensating behaviour as r -> infty. Though in the short
distance limit a dipole behaviour is present. We also demonstrate enhancement
in the propagator of the bosonic localizing Zwanziger ghost field when the one
loop Gribov gap equation is satisfied. The explicit form of the one loop gap
equation for the Gribov mass parameter is also computed in the MOM scheme and
the zero momentum value of the renormalization group invariant effective
coupling constant is shown to be the same value as that in the MSbar scheme.
