Title: The first moment of cusp form L-functions in weight aspect on average,
Abstract: 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.