Let be a primitive Dirichlet character of conductor and let us denote by the associated -series. It is well known that there exists a constant such that satisfies the following bound:

Recall that is said to be even or odd according to whether or . It has been proven by Ramaré that is possible when is even and when is odd. In the case , Ramaré, following the work of Louboutin, has already proposed an explicit improvement of the bound above. In this talk, we examine the harder case . We present a method that leads to a better value of when is even, .