The first moment of the symmetric-square L-function

We find a twisted first moment of L(sym²f,s) at any point s on the critical line, over a basis of weight k Hecke eigenforms f for the full modular group, as k→∞. As a corollary we show that given any point on the critical line and large enough even k, there exists an eigenform f of weight k such that L(sym²f,s) is nonvanishing at that point.