  Multi-dimensional state-integrals of products of Faddeev's quantum
dilogarithms arise frequently in Quantum Topology, quantum Teichm\"uller theory
and complex Chern--Simons theory. Using the quasi-periodicity property of the
quantum dilogarithm, we evaluate 1-dimensional state-integrals at rational
points and express the answer in terms of the Rogers dilogarithm, the cyclic
(quantum) dilogarithm and finite state-sums at roots of unity. We illustrate
our results with the evaluation of the state-integrals of the $4_1$, $5_2$ and
$(-2,3,7)$ pretzel knots at rational points.

Geometric Topology;High Energy Physics - Theory