Canonical divisor of an even-degree hyperelliptic curve
= Canonical divisor of an even-degree hyperelliptic curve
For the smooth curve $y^2=f(x)$ with square-free $f$ of degree $2n$, the differential $dx/y$ has divisor
$$
(n-2)(P_++P_-).
$$
Thus the canonical degree is $2n-4$, the genus is $n-1$, and $L(K)$ has basis $1,x,\ldots,x^{n-2}$.