  We prove two-sided estimates for the best (i.e., the smallest possible)
constant $\,c_n(\alpha)\,$ in the Markov inequality $$
  \|p_n'\|_{w_\alpha} \le c_n(\alpha) \|p_n\|_{w_\alpha}\,, \qquad p_n \in
{\cal P}_n\,. $$ Here, ${\cal P}_n$ stands for the set of algebraic polynomials
of degree $\le n$, $\,w_\alpha(x) := x^{\alpha}\,e^{-x}$, $\,\alpha > -1$, is
the Laguerre weight function, and $\|\cdot\|_{w_\alpha}$ is the associated
$L_2$-norm, $$
  \|f\|_{w_\alpha} = \left(\int_{0}^{\infty} |f(x)|^2
w_\alpha(x)\,dx\right)^{1/2}\,. $$ Our approach is based on the fact that
$\,c_n^{-2}(\alpha)\,$ equals the smallest zero of a polynomial $\,Q_n$,
orthogonal with respect to a measure supported on the positive axis and defined
by an explicit three-term recurrence relation. We employ computer algebra to
evaluate the seven lowest degree coefficients of $\,Q_n\,$ and to obtain
thereby bounds for $\,c_n(\alpha)$. This work is a continuation of a recent
paper [5], where estimates for $\,c_n(\alpha)\,$ were proven on the basis of
the four lowest degree coefficients of $\,Q_n$.
