  We present the order $\alpha_s^2$ contributions to the coefficient functions
corresponding to the longitudinal fragmentation function $F_L(x,Q^2)$. A
comparison with the leading order $\alpha_s$ result for $F_L(x,Q^2)$ shows that
the corrections are large and vary from 44\% to 67\% in the region $0.01 < x <
0.9$ at $Q^2=M_Z^2$. Our calculations also reveal that the ratio of the
longitudinal and total cross section $\sigma_L/\sigma_{\rm tot}$ amounts to
0.054. This number is very close to the most recent value obtained by the OPAL
collaboration which obtained $0.057\pm 0.005$.
