Transcendence measure for the values of the logarithmic derivative of the Bessel function $J_0$ at algebraic arguments

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\varepsilon>0$, there exists $c>0$ s