Let Q(z) be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke–Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue of $u_j (z)$. In this paper, we prove that there is a constant $ \gamma ( Q ) $ expressed as a certain Euler product associated to Q such that at least $ \gamma ( Q ) / 11 $ of the Rankin–Selberg special L-values $L (1/2+it_j, Q \otimes u_j)$ for $ t_j \leqslant T$ do not vanish as $T \rightarrow \infty $. Further, we show that the non-vanishing proportion is at least $\gamma ( Q ) \cdot (4\mu -3) / (4\mu +7) $ on the short interval $ |t_j - T| \leqslant T^{\mu } $ for any $3/4< \mu < 1$.
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