We compute from first principles the dispersion relation of a finite-temperature dipolar Bose gas near the roton instability. Using a field-theoretic Complex Langevin (CL) approach, we access experimentally relevant atom numbers. We determine the roton excitation gap as a function of the s-wave scattering length under the conditions of the experimental study by Petter et al. [Phys. Rev. Lett. 122, 183401 (2019)] using erbium atoms. Our results show good agreement with experimental measurements close to the instability, in contrast to predictions of the standard approximate field theory based on the extended Gross-Pitaevskii equation (eGPE). While the eGPE agrees with our results far from the instability, we find that it quantitatively fails as the roton softens, where the CL result lies between the GPE and eGPE predictions. This deviation reveals limitations of the (extended) mean-field description and establishes Complex Langevin as a powerful tool for quantitatively accurate simulations of many-body quantum systems where quantum fluctuations and temperature play an essential role.