Conjecture on the smallest positive zero of the Mittag–Leffler function with parameter beta equal to alpha

Let $E_{\alpha,\beta}$ denote the Mittag–Leffler function, and let $Z_{\alpha}(\alpha)$ be the smallest positive zero of $z\mapsto E_{\alpha,\alpha}(-z^\alpha)$.

Smallest-zero conjecture for $E_{\alpha,\alpha}$. For any $\alpha\in(1,2]$, $E_{\alpha,\alpha}(-z^\alpha)>0$ for all $z\in[0,2.93785]$. Moreover, as $\alpha\to1+$,

$$Z_{\alpha}(\alpha)=(c\ln(\alpha-1)+d)(1+o(1)),$$

with $c\approx-0.81$ and $d\approx2.25$, while as $\alpha\to2-$,

$$Z_{\alpha}(\alpha)=\pi+c(2-\alpha)+o(2-\alpha),$$

with $c\approx-0.81$.

These claims are inferred from numerical calculations and asymptotic fits; the two coefficients reported as approximately $-0.81$ may or may not be exactly equal, and no proof is provided.