Polarized and unpolarized neutron scattering was used to measure the wave vector-and frequency-dependent magnetic fluctuations in the normal state (from the superconducting transition temperature, T-c = 35 kelvin, up to 350 kelvin) of single crystals of La1.86Sr0.14CuO4. The peaks that dominate the fluctuations have amplitudes that decrease as T-2 and widths that increase in proportion to the thermal energy, k(B)T (where k(B) is Boltzmann’s constant), and energy transfer added in quadrature. The nearly singular fluctuations are consistent with a nearby quantum critical point.