simple give a list in the style property of the lines, like in

f_int_df = pandas.DataFrame(f_int_lst)
f_int_df['ds_pos'] = -F(q0, s_pos, alpha=alpha, gamma=gamma)
f_int_df.set_index('s')[['sum', 'ds_pos']].plot(
    figsize=(4,2), lw=3, grid=True, 
    style=['-', '--'], 
    title="$\propto\int_{q_0\wedge(q_0-s)}^\infty (1+a)^{-3} (1+a+s)^{-3}\, da$"
)

that gives (focus on the style=['-', '--'] line of code):