Multiple Dirichlet Series have been used as a tool for studying L-functions. For example, a Dirichlet series whose coefficients are themselves L-functions can be used to obtain mean value information using a Tauberian theorem. In the most interesting cases, the multiple Dirichlet series satisfies a group of functional equations. If it is finite, it may be isomorphic to a finite reflection group.
Weyl Group multiple Dirichlet series are a recently discovered class of such multiple Dirichlet series. Two theories of these objects are being developed, by Chinta and Gunnells, and by Brubaker, Bump and Friedberg. They are multiple Dirichlet series whose coefficients are constructed from Gauss sums. The coefficients are multiplicative, but these are not Euler products. Problems include completing these theories, and proving that they describe the same objects.
At the moment this seems most relevant to the encyclopedia and software parts of the project. Computation has been used in developing these theories but systematic tables do not seem to be an important goal at this time. This could change, perhaps within a year.
It was pointed out during the workshop that one natural parametrizing set for the coefficients of these Dirichlet series are the vertices of the crystal graph, an object that comes up in the representation theory of quantum groups. This increases the likelihood that software for this project would be of interest for a broader group of mathematicians.
http://sporadic.stanford.edu/bump/wmd.pdf
http://arxiv.org/abs/math/0703040
http://arxiv.org/abs/math/0612595
http://arxiv.org/abs/math/0604449
http://match.stanford.edu/bump/gelbart.pdf
http://annals.math.princeton.edu/issues/2006/FinalFiles/BrubakerBumpFriedbergHoffsteinProof.pdf