Determination of GL(3) Cusp Forms by Central Values of Quadratic Twisted L-Functions

Let phi and phi' be two GL(3) Hecke-Maass cusp forms. In this paper, we prove that phi = phi' or (phi) over tilde' if there exists a nonzero constant kappa such that L(1/2, phi circle times chi(8d)) = L(1/2, phi' circle times chi(8d)) for all positive odd square-free positive d. Here, (phi) over tilde' is dual form of phi' and chi(8d) is the quadratic character (8d/.). To prove this, we obtain asymptotic formulas for twisted 1st moment of central values of quadratic twisted L-functions on GL(3), which will have many other applications.