Reformulation of the Li criterion for the Selberg class

Let $F$ be a function in the Selberg class ${\mathcal S}$ and $a$ be a real number not equal to 1/2. Consider the sum $λ_{F}(n,a)=\sum_ρ\left[1-\left(\frac{ρ-a}{ρ+a-1}\right)^{n}\right],$ where $ρ$ runs over the non-trivial zeros of $F$. In this paper, we prove that the Riemann hypothesis is equivalent to the positivity of the "modified Li coefficient" $λ_{F}(n,a)$, for $n=1,2,..$ and $a<1/2$. Furthermore, we give an explicit arithmetic and asymptotic formula of these coefficients.

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