We present an extraction of the lowest three moments of the proton longitudinal structure function FL from world data between Q(2)=0.75 and 45 (GeV/c)(2). The availability of new FL data at low Bjorken x from HERA and at large x from Jefferson Lab allows the first determination of these moments over a large Q(2) range, relatively free from uncertainties associated with extrapolations into unmeasured regions. The moments are found to be underestimated by leading twist structure function parametrizations, especially for the higher moments, suggesting either the presence of significant higher twist effects in FL and/or a larger gluon distribution at high x.