Characterize when the evaluation formulas are explicit

Determine in which cases the evaluation formulas for the reducible irregular ${}_3F_2(1)$ hypergeometric motives can be made explicit, including the algebraic constants appearing in the decomposition into period terms.

Background

The main decomposition theorem proves that, under suitable reducibility and motivicity hypotheses, a normalized 3F2(1){}_3F_2(1) value is an algebraic linear combination of two periods, or equivalently two gamma-quotient-type quantities under the Gross–Deligne conjecture. The paper computes these algebraic coefficients in several examples, but leaves open a general criterion or method for determining them. The authors specifically note that resolving this question for HD(1/24,5/24)HD(1/24,5/24) would yield exact special values for the associated modular forms.

References

Question 2: In which cases can we make our evaluation formulas explicit?

— The Arithmetic of Reducible Rank 2 Hypergeometric Motives  (2609.28065 - Rosen, 23 Sep 2026) in Section 1, immediately following Question 1

We leave such a computation and proving the correct values to future work.

— The Arithmetic of Reducible Rank 2 Hypergeometric Motives  (2609.28065 - Rosen, 23 Sep 2026) in Section 5, subsection “Evaluation Formulas and $L$-Values,” paragraph following the proof of Theorem \ref{lvalirr} Part II