We propose the operation of textbf{LEvEL}, the Low-Energy Neutrino Experiment at the LHC, a neutrino detector near the Large Hadron Collider Beam Dump. Such a detector is capable of exploring an intense, low-energy neutrino flux and can measure neutrino cross sections that have previously never been observed. These cross sections can inform other future neutrino experiments, such as those aiming to observe neutrinos from supernovae, allowing such measurements to accomplish their fundamental physics goals. We perform detailed simulations to determine neutrino production at the LHC beam dump, as well as neutron and muon backgrounds. Measurements at a few to ten percent precision of neutrino-argon charged current and neutrino-nucleus coherent scattering cross sections are attainable with 100~ton-year and 1~ton-year exposures at LEvEL, respectively, concurrent with the operation of the High Luminosity LHC. We also estimate signal and backgrounds for an experiment exploiting the forward direction of the LHC beam dump, which could measure neutrinos above 100 GeV.
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