The first moment of cusp form L-functions in weight aspect on average

We study the asymptotic behaviour of the twisted first moment of central $L$-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier $\Delta$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $\Delta<1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.