Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?

Edit: Further cases of "fractional motives" as discussed in the article above (but with other weights) are expected to arise in quantum cohomology, say the experts. I wonder how the idea of such new motives may fit into the "usual" connections between motives, l-adic representations, periods and values of L-functions?

Edit: Conc. L-functions at non-integer values s, someone said that there is a quite old heuristical idea about it "as the dimension of an auxiliary affine space $A^s$ on which you multiply a given scheme over integers". Having either never read about that, or forgotten it: Do you know what it means and where to read more?