Given an abelian group with a divisor $D$ s.t. $L(D)$ gives projective embedding. Then $L(2D)$ is generated by degree 2 monomial elements of $L(D)$?

Question

Given an abelian group torus $A=\frac{C^n}{\Lambda}$ with a divisor $D$ s.t. $L(D)$ gives projective embedding.

$\textbf{Q:}$ Is $L(2D)$ generated by degree 2 monomial elements of $L(D)$? How do I see this.(I want to see the effect of Veronese embedding showing up somehow.) Riemann-Roch does not tell me this necessarily true? Is this true for large $n$ twist of $D$? Say $L(nD)$ for $n$ larger. Then $L(knD)$ is generated by degree $k$ monomial elements of $L(nD)$? How do I prove this? Hint only or reference only please.