On the first zeros of the Mittag-Leffler function 𝐸𝛼,𝛼(𝑧) for 𝛼>1

We investigate the first zeros (roots) of the Mittag-Leffler function by constructing and analyzing the eigenvalues of the associated fractional spectral problem. By extending special cases from previous studies, we establish a series of new results, including: (1) the function E α , α ( − z ) Eα ,α(-z) has no zeros within the disk centered at the origin with radius π 2 Γ ( α − 1 ) π ^2Γ (α -1) in the complex plane; (2) let λ ( α ) λ (α ) be the first zero (i.e., the zero with the smallest modulus) of E α , α ( − z ) Eα ,α(-z) , which always exists and is real, then the ratio Γ ( α − 1 ) λ ( α ) Γ (α -1)/λ (α ) is strictly decreasing over α ∈ ( 1 , ∞ ) α ∈ (1,∞ ) as a continuous function of α α . Moreover, extended results and theoretical insights are also presented.