Conjecture on the smallest positive zero of the Mittag–Leffler function with parameter beta equal to 1

Let $E_{\alpha,\beta}$ denote the Mittag–Leffler function, and let $Z_1(\alpha)$ be the smallest positive zero of $z\mapsto E_{\alpha,1}(-z^\alpha)$.

Smallest-zero conjecture for $E_{\alpha,1}$. For any $\alpha\in(1,2]$, $E_{\alpha,1}(-z^\alpha)>0$ for all $z\in[0,1.559]$. Moreover, as $\alpha\to1+$,

$$Z_1(\alpha)=(c\ln(\alpha-1)+d)(1+o(1)),$$

with $c\approx-0.275$ and $d\approx17.54$, while as $\alpha\to2-$,

$$Z_1(\alpha)=\frac{\pi}{2}+c(2-\alpha)+o(2-\alpha),$$

with $c\approx-0.12$.

The claim is based on numerical calculations and suspected asymptotic behavior shown in the paper's figures. The positivity bound and the two asymptotic expansions, including their numerical coefficients, remain unproved in the supplied text.