Let S={α: log_2(9^2 α-4+13)- log_2(5/2 · 3^2 α-4+1)=2}. Then the maximum value of β for which the equation x^2-2(Σ_α ∈ S α)^2 x+Σ_α ∈ S(α+1)^2 β=0…

Let $\displaystyle S=\left\{\alpha: \log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2\right\}$. Then the maximum value of $\displaystyle \beta$ for which the equation $\displaystyle x^2-2\left(\sum_{\alpha \in S} \alpha\right)^2 x+\sum_{\alpha \in S}(\alpha+1)^2 \beta=0$ has real roots, is $\displaystyle \_\_\_\_$.