Completed-L-function variant of the rationality conjecture

Let $\Delta(\mathcal{M})$ be the fundamental line, let $\theta_\infty:\Delta(\mathcal{M})_\infty\overset\simeq\longrightarrow F_\infty$ be the comparison isomorphism, and let $\Lambda^*(\mathcal{M},0)$ be the leading term of the completed $L$-function at $s=0$. Completed rationality conjecture. There exists $\zeta_f^*\in\Delta(\mathcal{M})$ such that

$$\theta_\infty(\zeta_f^*)=\Bigl((2\pi i)^{k/2}\Lambda^*(\mathcal{M},0)\Bigr)^{-1}$$

in $F_\infty^\times$. The source states that this is equivalent to the preceding rationality conjecture by the relation between $L^*(\mathcal{M},0)$ and $\Lambda^*(\mathcal{M},0)$.