occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Tuesday, March 18, 2008

### Riemann Rainbow

David Corfield brings our attention to an AIM press release about the discovery of a new L function. As the blurb explains, this new function $L(s)$ satisfies a degree 3 symmetric functional relation

$F(s) \equiv \frac{\sqrt{q}}{\pi^{3}} \Gamma (\frac{s}{2} + r_{1}) \Gamma (\frac{s}{2} + r_{2}) \Gamma (\frac{s}{2} + r_{3}) L(s) = F(1 - s)$

for some integer $q$, in contrast to the degree 1 behaviour of the Riemann zeta function (for which $q = 1$). Of course I immediately emailed Michael Rubinstein to ask for a reference on the actual values of these Langlands' parameters, as well as values for the first few known zeroes, which lie on the critical line. I eagerly await a reply, but my server may well be treated as a spam generator. In the meantime, Minhyong Kim has kindly provided helpful comments and links.