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Density-functional theory may be used to predict both the frequency and the dipole moment of the fundamental oscillations of molecular crystals. Suitably polarized photons at those frequencies excite such oscillations. Thus, in principle, terahertz spectroscopy may confirm the calculated fundamental modes of amino acids. However, reports to date have multiple shortcomings: (a) material of uncertain purity and morphology and diluted in a binder material is employed; (b) consequently, vibrations along all crystal axes are excited simultaneously; (c) data is restricted to room temperature, where resonances are broad and the background dominant; (d) comparison with theory has been unsatisfactory (in part because the theory assumes zero temperature). Here, we overcome all four obstacles, in reporting detailed low-temperature polarized THz spectra of single-crystal L-alanine, assigning vibrational modes using density-functional theory, and comparing the calculated dipole moment vector direction to the electric field polarization of the measured spectra. Our direct and detailed comparison of theory with experiment corrects previous mode assignments for L-alanine, and reveals unreported modes, previously obscured by closely-spaced spectral absorptions. The fundamental modes are thereby determined.