Let π be a cuspidal automorphic representation of GL3($\Bbb A$$\Bbb Q$), unramified at p and of cohomological type at infinity. We construct p-adic L-functions, which interpolate the critical values of L(π,s) and which satisfy a logarithmic growth condition. We obtain these functions as p-adic Mellin transforms of certain distributions μπ on $\Bbb Z$p* having values in some fixed number field and which are of moderate growth. In the p-ordinary case we obtain the bound |μπ(U)|p[les ]|μHaar(U)|p for open subsets U[les ] $\Bbb Z$p*, where μHaar denotes the invariant distribution on $\Bbb Z$p*.