Computation of Z3-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 λp(k) of the cyclotomic Zp-extension of k to be zero. The purpose of this paper is to study the case p = 3 of this result and give new examples of k with λ3(k) = 0, by using information on the initial layer of the cyclotomic Z3-extension of k.