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?
Edit: Some links: Yuri Manin had wondered if such things may exist (correct reference to Anderson's article on fractional "arithm. Hodge structures"), M. Marcolli wrote about such things in the context of "dimensional regularization" (and it's connection with log motives and motivic sheaves), Deligne extended representation theory to complex dimensions. It would be interesting to see how such speculations fit to Kedlaya's "fantasy in the key of p"...
Edit: new article by Matilde Marcoli: http://arxiv.org/abs/1310.2261