Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$ I'm trying to find a closed form (in terms of simpler functions) for the following hypergeometric function with a complex argument:
$$\mathcal{Q}=\,_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right).\tag1$$
I have a guess (supported by thousands of digits from numerical calculations) about its argument (phase), but no ideas about its absolute value yet:
$$\arg\mathcal{Q}\approx0.168669236010871306727578153...\stackrel?=\arccos\left(\frac{12+\sqrt{33}}{18}\right)\tag2$$
$$|\mathcal{Q}|\approx0.915170225773196416688677425...\tag3$$
Can you suggest any ideas how to prove the conjecture $(2)$? Is there a closed form for the absolute value?

As suggested by  gammatester in a comment, the conjecture $(2)$ is equivalent to
$$\arg\,B\left(\frac{8\,\sqrt{11}\,i-5}{27};\,\frac12,\frac13\right)\stackrel?=\frac\pi3,\tag4$$
where $B$ denotes the incomplete beta function.

Also, it can be shown that another equivalent form of the conjecture $(2)$ is
$$B\left(\frac19;\,\frac16,\frac13\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\,\Gamma\left(\frac13\right)}{2\,\sqrt\pi}.\tag5$$
 A: Not sure how to transform conjecture $(2)$ to $(5)$. However, $(5)$ is true.
Let $\;\displaystyle t = \frac{1}{1+y^3}\;$, we can
rewrite the integral $\;\displaystyle B\left(\frac19;\frac13,\frac16\right)\;$ as
$$
\int_0^{1/9} t^{-5/6} (1-t)^{-2/3} dt
= \int_\infty^2 (1+y^3)^{5/6}\left(\frac{1+y^3}{y^3}\right)^{2/3}\frac{-3y^2 dy}{(1+y^3)^2}
= 3 \int_2^\infty \frac{dy}{\sqrt{1+y^3}}
$$
Let $\omega = e^{\pi i/3}$ and $\mathbb{T} = \big\{\; m+n\omega : m, n \in \mathbb{Z}\;\big\}$ be the triangular lattice span by $1$ and $\omega$. Let $\wp(z)$ be the Weierstrass elliptic $\wp$ function with double poles on lattice $\mathbb{T}$:
$$\wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \mathbb{T} \setminus \{ 0 \}} \left(\frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}\right)$$
Let $\;\displaystyle\eta = \frac{\Gamma\left(\frac13\right)\Gamma\left(\frac16\right)}{\sqrt{3\pi}}\;$, it is known that $\wp(z)$ satisfies a differential equation of the form:
$$\wp'(z)^2 = 4 \wp(z)^3 - g_2 \wp(z) - g_3\quad\text{ where }\quad g_2 = 0 \;\text{ and }\;g_3 = \frac{\eta^6}{16}$$
Let $\;\displaystyle y(z) = -\frac{4}{\eta^2} \wp\left(\frac{iz}{\eta}\right)$. Using symmetry, it is not hard to see as $z$ varies from $0$ to $\frac{\eta}{\sqrt{3}}$ along the real axis, $y(z)$ remains real and positive, decreases from $\infty$ at $z = 0$ to $0$ at $z = \frac{\eta}{\sqrt{3}}$. 
In terms of $z$, we have:
$$\frac{dy}{\sqrt{1+y^3}} = 
\frac{-\frac{4i}{\eta^3}\wp'\left(\frac{iz}{\eta}\right)dz}{
\sqrt{1-\frac{64}{\eta^6}\wp\left(\frac{iz}{\eta}\right)^3}}
= 
\frac{-4i\wp'\left(\frac{iz}{\eta}\right)dz}{
\sqrt{16g_3-64\wp\left(\frac{iz}{\eta}\right)^3}}
= - dz
$$
From this, we get
$$B\left(\frac19;\frac13,\frac16\right) = -3 \left[ y^{-1}(\infty) - y^{-1}(2)\right]
= 3y^{-1}(2)$$
This allow us to simplify conjecture $(5)$ 
$$
B\left(\frac19;\frac13,\frac16\right) \stackrel{?}{=} 
\frac{\Gamma\left(\frac13\right)\Gamma\left(\frac16\right)}{2\sqrt{\pi}} = \frac{\sqrt{3}}{2}\eta
\iff y^{-1}(2) \stackrel{?}{=} \frac{\eta}{2\sqrt{3}}
\iff  \wp(\frac{i}{2\sqrt{3}}) \stackrel{?}{=} -\frac{\eta^2}{2}
$$
Let $u = \frac{i}{2\sqrt{3}}$. Since $\wp(2u) = \wp\left(\frac{i}{\sqrt{3}}\right) = 0$, we can use the duplication formula for Weierstrass elliptic $\wp$ function to obtain
$$\begin{align}
& 0 = \wp(2u) = \frac14\left(\frac{(6\wp(u)^2-\frac12 g_2)^2}{4\wp(u)^3-g_2\wp(u)-g_3}\right) -2\wp(u) \\
\iff & \wp(u)\left(\frac{\wp(u)^3 + 2g_3}{4\wp(u)^3-g_3}\right) = 0\\
\implies & \wp(u) = (-2g_3)^{1/3} = \left(-\frac{\eta^6}{8}\right)^{1/3} = -\frac{\eta^2}{2}
\end{align}$$
i.e. the conjecture $(5)$ is true.
