Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to

$$ \chi(X, L)=\frac{1}{2}(L\cdot L)+2 $$

which can be rewritten as $$ h^0(X, L)+h^0(X, L^\ast)=\frac{1}{2}(L\cdot L)+2+h^1(X, L) $$

(I use Serre's duality to identify $H^2(X, L)$ and $H^0(X, L^\ast)^\ast$)

Assume $(L\cdot L)\geq 2$, then the RHS is bigger or equal than 2, so $H^0(X, L) \neq 0$ or $H^0(X, L^\ast)\neq 0$.

I guess the converse statement is not true, that is: there exists a line bundle $L$ such that $H^0(X, L)\neq 0$ but $(L\cdot L)<-2$. But I don't manage to give an example. Could anybody help me?