I will outline a strategy, based on the arithmetic of hypergeometric functions, to build Landau-Ginzburg (LG) mirrors for Fano anticanonical weighted complete intersections. The feature which makes these varieties especially interesting is that, if the ambient space has all weights greater than one, they have empty anticanonical linear system and therefore fall out of any known mirror construction. I will show how to use the strategy to produce LG mirrors for the series of anticanonical del Pezzo surfaces constructed by Johnson and Kollár.