Application of the Riemann-Roch theorem

Question

If C is a quadric hyperelliptic curve ($g(C)=3$ and the canonical line bundle is very ample) contained in the two dimensional complex projective space and $K_C$ is the canonical line bundle of $\mathbb{P}^2$ restrected to my curve.
I take four distinct points on $C$ named $p_1,\ p_2,\ p_3,\ p_4 $ so $p_1+ p_2+p_3+p_4 \in K_C= |\mathcal{O}_{\mathbb{P^2}}(1)| $. Than i take another points different from the previous, named $q$ and the divisor on $C$ $\ D=p_1+ p_2+p_3+q$.
I want to compute
$h^o(D)$.
So using Riemann Roch theorem i can write:
$h^o(D)=h^1(D)+deg(D)-g+1=h^1(D)+4-3+1=h^1(D)+2$.
Now i've used Serre Duality and i get:
$h^1(D)=h^0(K_C-D)$. Is there a way to say that $h^0(K_C-D)=0$ for example using some vanishing theorem?