Non-vanishing of the symmetric square $L$-function at the central point

Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric square $L$-function does not vanish at the central point of its critical strip. We note that our proportion is the same as that found by other authors for other families of $L$-functions also having symplectic symmetry type.