This thesis examines the relationship between the L-series of an elliptic curve evaluated at s = 2 and the image of the regulator map when the curve is defined over a real quadratic field with narrow class number one, thus providing numerical evidence for Beilinson's conjecture. In doing so it provides a practical formula for calculating the L-series for modular elliptic curves over real quadratic fields, and in outline for more general totally real fields, and also provides numerical evidence for the generalization of the Taniyarna-Weil-Shimura conjecture to real quadratic fields.