For f a cuspidal modular form for the group Γ0(N) of integral or half-integral weight, N a multiple of 4 in case the weight is half-integral, we study the zeros of the L-function attached to f twisted by an additive character e2πinpq with pq∈Q. We prove that for certain f and pq∈Q, the additively twisted L-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses automorphic distributions to prove the result.