  Generalized Baxter's relations on the transfer-matrices (also known as
Baxter's TQ relations) are constructed and proved for an arbitrary untwisted
quantum affine algebra. Moreover, we interpret them as relations in the
Grothendieck ring of the category O introduced by Jimbo and the second author
in arXiv:1104.1891 involving infinite-dimensional representations constructed
in arXiv:1104.1891, which we call here "prefundamental". We define the
transfer-matrices associated to the prefundamental representations and prove
that their eigenvalues on any finite-dimensional representation are polynomials
up to a universal factor. These polynomials are the analogues of the celebrated
Baxter polynomials. Combining these two results, we express the spectra of the
transfer-matrices in the general quantum integrable systems associated to an
arbitrary untwisted quantum affine algebra in terms of our generalized Baxter
polynomials. This proves a conjecture of Reshetikhin and the first author
formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe
Ansatz equations for all untwisted quantum affine algebras.
