We present a special value formula for the L-function of a cuspidal modular form on the upper half plane, twisted by a Grossencharacter of a real quadratic field. The geometric objects entering the formula are certain geodesic cycles on the modular curve X_0(N), which are associated with the real quadratic field. We discuss two applications of the formula to the study of such geodesics. On the analytic side, we show that individual "long" geodesics are equidistributed on X_0(N). On the arithmetic side, assuming that the modular form is associated with an elliptic curve E over the rationals, we show that the order of the Shafarevich-Tate group of E over the quadratic field can be interpreted solely in terms of the homology class of these geodesics.