We present an analysis of the recently discovered blue L dwarf SDSS J141624.08+134826.7. We extend the spectral coverage of its published spectrum to ~4 microns by obtaining a low-resolution L band spectrum with SpeX on the NASA IRTF. The spectrum exhibits a tentative weak CH4 absorption feature at 3.3 microns but is otherwise featureless. We derive the atmospheric parameters of SDSS J141624.08+134826.7 by comparing its 0.7-4.0 micron spectrum to the atmospheric models of Marley and Saumon which include the effects of both condensate cloud formation and non-equilibrium chemistry due to vertical mixing and find the best fitting model has Teff=1700 K, log g=5.5 [cm s-2], fsed=4, and Kzz=10^4 cm2 s-1. The derived effective temperature is significantly cooler than previously estimated but we confirm the suggestion by Bowler et al. that the peculiar spectrum of SDSS J141624.08+134826.7 is primarily a result of thin condensate clouds. In addition, we find strong evidence of vertical mixing in the atmosphere of SDSS J141624.08+134826.7 based on the absence of the deep 3.3 micron CH4 absorption band predicted by models computed in chemical equilibrium. This result suggests that observations of blue L dwarfs are an appealing way to quantitatively estimate the vigor of mixing in the atmospheres of L dwarfs because of the dramatic impact such mixing has on the strength of the 3.3 micron CH4 band in the emergent spectra of L dwarfs with thin condensate clouds.