Slightly improved zero-free half-planes for the quasi-Riemann hypothesis

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [1]. In its proof, there are two chosen parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$. In this note, via varying $