On a choice of the mollified function in the Levinson-Conrey method

Motivated by a functional property of the Riemann zeta function, we consider a new form of the mollified function in the Levinson-Conrey method. As an application, we give the following slight improvement of Feng's result: assuming Feng's condition on the lengths of the mollifier at least 41.2948% of the zeros of the Riemann zeta function are on the critical line. The construction may lead to further improvements as one increases the number of terms in Feng's mollifier.