Cohomology of Pure Braid Groups of exceptional cases

Consider the ring R:=Q[τ,τ−1] of Laurent polynomials in the variable τ. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ. In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We compute this cohomology for the cases I2(m), H3, H4, F4 and An with 1⩽n⩽7.