We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth Riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, admits the standard metric g0 as a critical point, although it does not achieve the conjectured global minimum: we show that for each tangent direction to the space of metrics at g0, there exists a variation by metrics corresponding to this direction along which the systolic area can only increase.