Application of the Castelnuovo's contraction criterion (CCC)

Question

I've seen the castelnuovo's contraction criterion that says:
if $E \in S$ is a $(-1)$ curve contained in a complex projective algebraic surface than $E$ is an exceptional curve.
Then there is an application of this theorem, in particular an application using the proof, to the cubic scrool in $\mathbb{P^4}$.
Let $S \in \mathbb{P^4}$ the cubic scroll in $ \mathbb{P^4}$. We know that the picard group of $S$ is a free abelian group of rank $2$ so $Pic(S)=<l,f>$. So every divisor in $Pic(S)$ is equivalen to a linear combination of $l$ and $f$.
In particular the hyperplane bundle can be written as $L \tilde{}l+2f $, and the canonical line bundle as $K_S \tilde{}-2l-3f $.
By kawamata vanishing theorem we get $h^1(L)=0$. Following the proof of the CCC we get that $L'=L+l$ is spanned so the morphism $\psi: S \rightarrow \mathbb{P^a}$ is an embedding out of $\psi(l)=p_0$. I want to compute the dimensin $a$ of the projective space seen before.
I know that $a = h^0 ( L^{'} )-1=h^0 (L)$ but i don't know how to compute $h^0 (L)$. Is there a way to do this?