Let \(f\) be a primititve modular form of weight \(k\) and level \(q\) given by the Fourier expansion \[ f(x)= \sum^\infty_{n=1} n^{(k-1)/2} \lambda(n) e^{2\pi inz}, \] and let \(L_d(s)\) denote the integral function such that \[ L_d(s)= \sum^\infty_{n=1} \lambda(n) \chi_d(n) n^{-s} \] for \(\text{Re\,}s> 1\), where \(\chi_d\) is a real primitive character modulo \(d\), \(d\in A\), and \(A\) stands for the set of the square-free positive integers co-prime to \(2q\). Further, let \[ \Lambda_d(s)= (dq^{1/2}/2\pi)^{s- 1/2} \Gamma(s+ (k-1)/2) L_d(s) \] be the completed \(L\)-function. The author proves the following asymptotic formula for the \(l\)-th derivative of \(\Lambda\): \[ \sum_{d\in A} r(d)\Lambda^{(l)}_d(1/2) F(d/Y)= YQ_l(\log Y)+ O(Y(\log Y)^{15/2}), \] where \(r(d)\) denotes the number of representations of \(d\) as a sum of two squares, \(F\) is a smooth compactly supported function, and \(Q_l\) is a polynomial of degree \(l\).
In the last section of his paper, the author briefly indicates how one could obtain a better estimate of the remainder in his asymptotic formula. This work is inspired by the author’s investigations of the elliptic fibrations of the shape \((t^2+ 1)y^2= g(x)\), where \(g(x)\) is a cubic polynomial.