The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition $\lambda=\{\lambda_1\geq \lambda_2\geq \lambda_3\geq 0\}$ of non-negative integers, we calculate the Euler characteristic of the local system $\Bbb V_\lambda$ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some $\lambda$ of low degree, we make a guess for the motivic Euler characteristic of $\Bbb V_\lambda$ using the counting of curves over finite fields.