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Through use of multiband (U, B, R, I) photometry we have isolated highredshift (3.0<z<3.5) galaxy candidates in a survey of 1.27 deg^2 to R = 21.25and a survey of 0.02 deg^2 to R = 23.5. Our pool of candidates constrains thenature of the 3.0 < z < 3.5 luminosity function over the range L* < L < 100 L*,if we grant a similar level of completeness to these data as for very faintsamples (to R = 25.5) selected in a similar fashion. Our constraints agree withthe high redshift sky density at R = 20.5 estimated from Yee et al.'s (1996)serendipitous discovery of a bright, z = 2.7 galaxy, as well as the density atR ~ 23 by Steidel et al. (1996b). We strongly rule out -- by more than twoorders of magnitude at M(R) = -25 -- the L > L* luminosity function for z = 3-5galaxies obtained by a photometric redshift analysis of the Hubble Deep Field(HDF) by Gwyn & Hartwick (1996). Our results at R ~ 23 are more consistent withthe photometric redshift analysis of the faint HDF galaxies by Sawicki & Yee(1996), but our present upper limits at the brightest magnitudes (R < 21.5,M(R) < -24) allow more generous volume densities of these super-L* galaxies.Comment: Accepted for publication in ApJ Letters; 14 pages Latex, including 3  figure

The Luminosity Function for [ITAL]L[/ITAL] ] [ITAL]L[/ITAL]* Galaxies at [CLC][ITAL]z[/ITAL][/CLC] ] 3