These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of, $${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}\tag1$$ The function $\eta(\tau)$ below is the Dedekind eta function.
I. Case $a=\tfrac13$
Conjecture: There are an infinitely many algebraic numbers $\alpha, \beta$ such that $$H_1(\tau) =\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-\alpha \big)=\beta$$ given by, $$\alpha = \frac1{4\sqrt{27}}\big(\lambda^3-\sqrt{27}\,\lambda^{-3}\big)^2$$ where $\lambda=\large{\frac{\eta(\frac{\tau+1}3)}{\eta(\tau)}}$ and $\tau=\frac{1+N\sqrt{-3}}2$ for any integer $N>1$. Examples: $$H_1\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-4 \big)=\tfrac3{5^{5/6}}$$ $$H_1\big(\tfrac{1+7\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27 \big)=\tfrac47$$ More details here.
II. Case $a=\tfrac14$
Conjecture: There are an infinitely many algebraic numbers $\alpha, \beta$ such that $$H_2(\tau) =\,_2F_1\big(\tfrac14,\tfrac14;\tfrac34;-\alpha \big)=\beta$$ given by, $$\alpha = \frac1{4\sqrt{64}}\big(\lambda^6-\sqrt{64}\,\lambda^{-6}\big)^2$$ where $\lambda=\large{\frac{\sqrt2\,\eta(2\tau)}{\zeta_{48}\,\eta(\tau)}},\,$ $\zeta_{48} = e^{2\pi i /48},$ and $\tau=\frac{1+N\sqrt{-1}}2$ for any integer $N>1$. Examples: $$H_2\big(\tfrac{1+3\sqrt{-1}}2)=\,_2F_1\big(\tfrac14,\tfrac14;\tfrac34;-3 \big)=\tfrac2{3^{3/4}}$$ $$H_2\big(\tfrac{1+5\sqrt{-1}}2)=\,_2F_1\big(\tfrac14,\tfrac14;\tfrac34;-80 \big)=\tfrac35$$ More details here.
III. Case $a=\tfrac16$
Conjecture: There are an infinitely many algebraic numbers $\alpha, \beta$ such that $$H_3(\tau) =\,_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-\alpha \big)=\beta$$ with $\alpha$ as a root of the quadratic, $$16\cdot432\,\alpha(1+\alpha)=-j(\tau)$$ with j-function $j(\tau)$ and $\tau = \frac{1+N\sqrt{-3}}{2}$ for any integer $N>1$. Examples: $$H_3\big(\tfrac{1+3\sqrt{-3}}2)=\,_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-\tfrac{125}3 \big)=\tfrac2{3^{5/6}}$$ $$H_3\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9 \big)=\tfrac3{5^{5/6}}\phi^{-1}$$ More details here.
Q: Are the three related conjectures true? And how to prove them? (Bounty is about to expire for the first one.)
