A global cohomological formula for the square class of the central value of a symplectic L-function

In the 70s Deligne gave a topological formula for the local epsilon factors attached to an orthogonal representation.
We consider the case of a symplectic representation and present a conjecture giving a topological formula for a finer invariant, the square class of its central value.
We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds and give a sketch of the proofs.
This is joint work in progress with Akshay Venkatesh.