  We calculate the third coefficient of the lattice $\beta$ function in pure
Yang-Mills theory. We make use of a computer code for solving perturbation
theory analytically on the lattice. We compute the divergent integrals by using
a method based on a Taylor expansion of the integrand in powers of the external
momenta in $4 - \epsilon$ dimensions. Our results are in agreement with a
previous calculation by M. L\"uscher and P. Weisz where the authors used a
different technique. We also show how this new coefficient modifies the scaling
function on the lattice in both the standard and energy schemes. In particular
we show that asymptotic scaling is extremely well achieved in the energy
scheme.
