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We analyse some peculiar properties of the function of the Mittag-Leffler(M-L) type, $e_\alpha(t):= E_\alpha(-t^\alpha)$ for $0 <\alpha < 1$ and $t >0$, which is known to be completely monotone (CM) with a non negative spectrumof frequencies and times, suitable to model fractional relaxation processes. Wefirst note that these two spectra coincide so providing a universal scalingproperty of this function. Furthermore, we consider the problem ofapproximating our M-L function with simpler CM functions for small and largetimes. We provide two different sets of elementary CM functions that areasymptotically equivalent to $e_\alpha(t)$ as $t \to 0$ and $t \to \infty$.

On some properties of the Mittag-Leffler function $E_\alpha(-t^\alpha)$, completely monotone for $t > 0$ with $0 < \alpha < 1$