In the present work, we provide the general expression of the normalized centered moments of the Fréchet extreme-value distribution. In order to try to represent a set of data corresponding to rare events by a Fréchet distribution, it is important to be able to determine its characteristic parameter $α$. Such a parameter can be deduced from the variance (proportional to the square of the Full Width at Half Maximum) of the studied distribution. However, the corresponding equation requires a numerical resolution. We propose two simple estimates of $α$ from the knowledge of the variance, based on the Laurent series of the Gamma function. The most accurate expression involves the Apéry constant.