Entry 35

This continues Entry 34. Similar to its more famous cousin, Euler's prime-generating polynomial, the formula $$P(n)=6n^2-6n+31$$ yields consecutive prime values from $n=0\to29$. Solving for $P(n)=0$, one gets $\tau=\frac{6+\sqrt{-708}}{12}=\frac{3+\sqrt{-177}}{6}$. Plugging this into $$\beta(\tau)=\Big (\left(\tfrac{\eta(2\tau)\,\eta(3\tau)}{\eta(\tau)\,\eta(6\tau)}\right)^6-\left(\tfrac{\eta(\tau)\,\eta(6\tau)}{\eta(2\tau)\,\eta(3\tau)}\right)^6\Big)^2$$ with Dedekind eta function $\eta(\tau)$, we find that it exactly yields an integer $$\beta\big(\tfrac{3+\sqrt{-177}}{6}\big)=-1060^2$$ such that $$e^{2\pi/6\sqrt{177}} = 1060^2+9.999992\dots$$ and the level-6 Ramanujan-Sato series $$\frac{1}{\pi}=\frac{1}{265\cdot1060}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k\tbinom{k}{j}^3\,\frac{177\cdot7038k+89418}{(-1060^2)^k}$$ with the binomial coefficient $\binom{n}{k}$.