  For a partition $lambda=\{lambda_1 \geq \lambda_2 \geq \lambda_3 \}$ of
non-negative integers, we calculate the Euler characteristic of the local
system $V_{\lambda}$ 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 $V_{\lambda}$ using counting of curves
over finite fields.
