This research is in the general area of the so-called Langlands Program, a subject that lies on the bridge between Number Theory and Representation Theory in Modern Analysis. More specifically one attaches an L-function to a cuspidal automorphic representation of the group of two by two matrices over the rational adele ring, and one also attaches an L-function to an irreducible representation of the galois group of the algebraic closure of the rational numbers. The main question to be investigated on this project is when these two L-function coincide. A new method has been devised to answer this question and this method will be pursued. Two different types of functions that contain a great deal of number theoretic information will be investigated. When they can be shown to coincide one can obtain much important arithmetic information about the more obscure one from the better understood one. Producing these formulas is the focus of this research.
