  Right adjoints for the forgetful functors on $\lambda$-rings and bi-rings are
applied to motivic measures and their zeta functions on the Grothendieck ring
of $\mathbb{F}_1$-varieties in the sense of Lorscheid and Lopez-Pena (torified
schemes). This leads us to a specific subring of $\mathbb{W}(\mathbb{Z})$,
properly containing Almkvist's ring $\mathbb{W}_0(\mathbb{Z})$, which might be
a natural receptacle for all local factors of completed zeta functions.
