Non-vanishing of the first derivative of GL(3) ×GL(2) L-functions

Let f be a fixed self-dual Hecke–Maass cusp form for SL(3, ℤ) and {μj} be an orthogonal basis of odd Hecke–Maass cusp forms for SL(2, ℤ). We prove an asymptotic formula for the average of the first derivative of the Rankin–Selberg L-function of f and μj at the center point s = 1 2. This implies the non-vanishing results for the first derivative of these L-functions at the center point s = 1 2.