Higher order approximations to the longitudinal structure function FL from the parametrization of F2 based on the Laplace transformation

We describe the determination of the longitudinal structure function FL at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) approximations, using Laplace transform techniques, into the parametrization of F2(x,Q2) and its derivative with respect to lnQ2 at low values of the Bjorken variable x. The obtained results are comparable with others by considering the effect of the charm quark mass to the longitudinal structure function, which leads to rescaling the variable for nf=4. Numerical calculations and comparison with H1 data demonstrate that the suggested method provides reliable FL(x,Q2) at small x in a wide range of Q2 values and can be applied as well in analyses of ultrahigh energy processes with cosmic neutrinos. The obtained longitudinal structure functions with and without the LHeC simulated uncertainties [CERN-ACC-Note-2020-0002, P. Agostini (LHeC Collaboration and FCC-he Study Group), J. Phys. G 48, 110501 (2021).] are compared with the H1 Collaboration data [V. Andreev et al. (H1 Collaboration), Eur. Phys. J. C 74, 2814 (2014) and F. D. Aaron et al. (H1 Collaboration), Eur. Phys. J. C 71, 1579 (2011)] and with the results from the CT18 [T.-J. Hou et al., Phys. Rev. D 103, 014013 (2021)] parametrization model at NLO and NNLO approximations.