Editing L-function (section)

== Conjectural information ==

One can list characteristics of known examples of ''L''-functions that one would wish to see generalized:

* location of zeros and poles;
* [[functional equation (L-function)|functional equation]], with respect to some vertical line Re(''s'') = constant;
* interesting values at integers related to quantities from [[algebraic K-theory|algebraic ''K''-theory]].

Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta function connects through its values at positive even integers (and negative odd integers) to the [[Bernoulli numbers]], one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for [[p-adic L-function|''p''-adic ''L''-function]]s, which describe certain [[Galois module]]s.

The statistics of the zero distributions are of interest  because of their connection to problems like the generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with [[random matrix]] theory and [[quantum chaos]] are also of interest. The fractal structure of the distributions has been studied using [[rescaled range]] analysis.<ref name="Shanker">{{cite journal|author=O. Shanker|year=2006|title=Random matrices, generalized zeta functions and self-similarity of zero distributions|journal=J. Phys. A: Math. Gen.|volume=39|issue=45 |pages=13983–13997    | doi = 10.1088/0305-4470/39/45/008|bibcode=2006JPhA...3913983S|s2cid=54958644 }}</ref> The [[self-similarity]] of the zero distribution is quite remarkable, and is characterized by a large [[fractal dimension]] of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the [[Riemann zeta function]], and also for the zeros of other ''L''-functions of different orders and conductors.