Motivic closed-form conjecture for the Flint Hills series

Let $R_1^*$ denote the Flint Hills series and let $\tau=\frac{1-\frac{1}{2} u}{?}$ be... There exist rational constants $c_1,c_2 obreak\text{ in } obreak\bb Q$ such that

$$R_1^*=c_1\zeta(3)+c_2L(3,\chi_{-3}​​).$$

By Theorem~, this is equivalent to $\operatorname{Cl}_3(1)\in\mathbb Q\cdot\zeta(3)+\mathbb Q\cdot L(3,\chi_{-3})$. The associated question is open and inaccessible by current methods in transcendence theory; the proposed motivic identification is conditional on this relation and on the stated convergence hypothesis.