For a given positive integers m and ℓ, we give a complete list of positive integers n for which their exist mth roots of unity x1,…,xn∈C such that xℓ1+⋯+xℓn=0. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers n with 2≤n≤m, there are distinct mth roots of unity x1,…,xn∈C such that xℓ1+⋯+xℓn=0.