Symmetry conjecture for the special values of $L_{+}$

Let $L_{+}$ be the function appearing in the paper, and let $\mathscr{C}$ be its associated constant. Symmetry conjecture. The values $L_{+}(2k)$ satisfy the following symmetry:

$$L_{+}(-2k) = \frac{L_{+}(2k)}{(2\pi i\mathscr{C})^{2k}},\qquad k\in\mathbb{Z}.$$

This symmetry is conjectured for the special values at negative even integers and has been verified numerically to very high precision.