Special-value conjecture for Ramanujan–Sato hypergeometric series and weight-3 modular forms

For a parameter tuple ${\bf b}$, let $F_{0,{\bf b}}(\lambda)$ denote the hypergeometric function used in the paper, and let $L(f,s)$ be the $L$-function of a weight-$3$ modular form $f$. The relevant modular forms are identified by their LMFDB labels.

Special-value conjecture.

$$\begin{aligned} \pi^2 F_{0,(\frac12,\frac{1}{6},\frac{5}{6})}\left(\frac{27}{125}\right)&=\frac{25}{2}L(f_{800.3.g.a},2),\\ \pi^2 F_{0,(\frac12,\frac{1}{6},\frac{5}{6})}\left(\frac{4}{125}\right)&=\frac{25}{4}L(f_{300.3.g.b},2),\\ \pi^2 F_{0,(\frac12,\frac{1}{4},\frac{3}{4})}\left(\frac{1}{9}\right)&=12L(f_{24.3.h.a},2),\\ \pi^2 F_{0,(\frac12,\frac{1}{3},\frac{2}{3})}\left(\frac{1}{2}\right)&=9L(f_{24.3.h.b},2),\\ \pi^2 F_{0,(\frac12,\frac{1}{3},\frac{2}{3})}\left(\frac{4}{125}\right)&=\frac{45}{4}L(f_{15.3.d.b},2). \end{aligned}$$

These identities are motivated by numerical evidence and by known evaluations of special hypergeometric values in terms of $L$-values of CM modular forms, but their general derivation is not established here.