Non-vanishing of derivatives of L-functions associated to cusp forms of half-integral weight in the plus space

We show a non-vanishing result for the derivatives of L-functions associated to cuspidal Hecke eigenforms of half-integral weight in plus space. In particular, we show that for large weights, ∑j=1d1⟨fk,j,fk,j⟩dndsn[L∗(fk,j|W4,s)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^{d}\frac{1}{\langle f_{k,j}, f_{k,j} \rangle }\frac{d^n}{ds^n}[L^*(f_{k,j}|W_4,s)] \end{aligned}$$\end{document}does not vanish at any point s=σ+it0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=\sigma +it_0$$\end{document} with t=t0,k/2-1/4<σ<k/2+3/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=t_0,k/2-1/4<\sigma <k/2+3/4$$\end{document}.