The longitudinal structure function at the third order

We compute the complete third-order contributions to the coefficient functions for the longitudinal structure function FL, thus completing the next-to-next-to-leading order (NNLO) description of unpolarized electromagnetic deep-inelastic scattering in massless perturbative QCD. Our exact results agree with determinations of low even-integer Mellin moments and of the leading small-x terms in the flavour-singlet sector. In this Letter we present compact and accurate parametrizations of the results and illustrate the numerical impact of the NNLO corrections.