Through use of multiband (U, B_J, R_F, I_N) photometry we have isolated high redshift (3.0 < z < 3.5) galaxy candidates in a survey of 1.27 deg^2 to R_F = 21.25 and a survey of 0.02 deg^2 to R_F = 23.5. Our pool of candidates constrains the nature of the 3.0 < z < 3.5 luminosity function over the range L* <~ L <~ 100 L*, if we grant a similar level of reliability to these data as has been found in very faint samples selected in a similar fashion. In support of our constraints, we find agreement with the high redshift sky density at R_F = 20.5 estimated from Yee et al.'s (1996) serendipitous discovery of such a bright, z = 2.7 galaxy, as well as the sky density at R_F ~ 23 obtained in the deep survey by Steidel et al. (1996). We strongly rule out -- by more than two orders of magnitude at M_{R_F} = -25 -- the L > L* luminosity function for z = 3-5 galaxies obtained by a photometric redshift analysis of the Hubble Deep Field (HDF) by Gwyn & Hartwick (1996). Our results at R_F ~ 23$ are more consistent with the photometric redshift analysis of the faint HDF galaxies by Sawicki & Yee (1996). However, our present upper limits at the brightest magnitudes (R_F < 21.5, M_{R_F} < -24) allow more generous volume densities of these super-L* galaxies.