Kirti Joshi's <a href="http://arxiv.org/abs/1005.3008" title="arxiv">musings</a> 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 <a href="http://arxiv.org/abs/math/0502016" title="arxiv">wondered</a> if such things may exist (correct reference to <a href="http://archive.numdam.org/ARCHIVE/CM/CM_1986__57_2/CM_1986__57_2_153_0/CM_1986__57_2_153_0.pdf" title="numdam">Anderson's article</a> on fractional "arithm. Hodge structures"), M. Marcolli <a href="http://arxiv.org/abs/0804.4824" title="arxiv">wrote</a> about such things in the context of "dimensional regularization" (and it's connection with log motives and motivic sheaves), Deligne <a href="http://www.math.ias.edu/files/deligne/Symetrique.pdf" title="pdf">extended</a> representation theory to complex dimensions. It would be interesting to see how such speculations fit to Kedlaya's <a href="http://math.mit.edu/~kedlaya/papers/nagoya2010.pdf" title="slides">"fantasy in the key of p"</a>... 

Edit: new article by Matilde Marcoli: http://arxiv.org/abs/1310.2261