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A new mathematical object, an elusive cousin of the Riemann zeta-function, was revealed to great acclaim recently at the American Institute of Mathematics. Ce Bian and Andrew Booker from the University of Bristol showed the first example of a third degree transcendental L-function.

Functional equations shed light on the properties of those functions that satisfy them, and for L-functions F(s) the functional equation is:   where q is an integer called the level, d is the degree, and the numbers  are Langland's parameters. is an analytic continuation of the factorial function  that is valid not only for integers but all complex numbers. There are two types of L-functions: algebraic and transcendental. These are classified according to their degree. If the Langland's parameters are rational or algebraic (that is, are complex numbers that are roots of non-zero polynomials with rational coefficients), then the L-function is algebraic. If these numbers are transcendental (that is, non-algebraic, such as  or  , then the L-function is transcendental. The Riemann zeta-function is the L-function where the level is 1, the degree is 1 and the Langland's parameters are 0 — that is, a first degree algebraic L-function. The Bristol researchers showed the first example of a third degree transcendental L-function.

wo researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function.

"This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.

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