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 \(\binom{n}{k}\).