Let $X$ be a symmetric L\'{e}vy process in $R^d, d=2,3.$ We assume that $X$ has independent $\alpha _j -$stable components, $1 < \alpha _d \leq \cdots \leq \alpha _1 < 2$ (a process with stable components, by Pruitt-Taylor (1969a)), or more generally that $X$ is $d$-dimmensiomally self-similar with similarity exponents $H_j, H_j=1/ \alpha _j$ (a dilation-stable process, by Kunita (1993)). Let a given integer $k \geq 2$ be such that $k(H-1) < H, H = \sum ^d _{j=1} H_j$. We prove that the set of $k-$multiple points $E_k$ is almost surely of Hausdorff dimension \[ \dim E_k = \min (\frac{k-(k-1)H}{H_1}, d- \frac{k(H-1)}{H_d}). \] In the stable components case, the above formula was proved by Hendricks (1974) for $d=2$ and suspected by him for $d=3$.