Computation of Z(3)-invariants of real quadratic fields

Let k be a real quadratic field and p an odd prime number which splits in k. In a previous work, the author gave a sufficient condition for the Iwasawa invariant lambda(p)(k) of the cyclotomic Z(p)-extension of k to be zero. The purpose of tills paper is to study the case p = 3 of this result and give new examples of k with lambda(3)(k) = 0, by using information on the initial layer of the cyclotomic Z(3)-extension of k.