Hooley's Riemann hypothesis for cubic six-variable Hasse–Weil L-functions

Let $\mathfrak g({\mathbf x})=x_1^3+\cdots+x_6^3$, let $\Delta({\mathbf m})$ be the associated discriminant, and let $\xi({\mathbf m};s)$ be the completed modified Hasse–Weil $L$-function defined from the point-counting errors of the projective variety $\mathfrak g({\mathbf x})={\mathbf m}\cdot{\mathbf x}=0$. Suppose that

$$\Delta({\mathbf m})\ne 0.$$

HRH. The function $\xi({\mathbf m};s)$ has a meromorphic continuation to $\mathbb C$ of finite order, with only possible poles at $s=\frac{3}{2}$ and $s=\frac{5}{2}$; there is a sign $w({\mathbf m})=\pm1$ such that

$$\xi({\mathbf m};s)=w({\mathbf m})\xi({\mathbf m};4-s);$$

and, when $\operatorname{Re}(s)\ne2$, one has $\xi({\mathbf m};s)\ne0$. This hypothesis would provide the conditional treatment of the $(l,m,n)=(5,5,2)$ systems of type D, for which the paper does not give an unconditional treatment.