Is there an explicit expression for the imaginary part of some nontrivial zero of zeta, in terms of wellknown constants, such as say $\gamma$ or $\pi$ say ?

$\begingroup$ Clearly there needs some amount of rigorisation here. There are infinite series with rational values that sum to the zeroes. There are a number of other infinite processes that may be considered fairly well known. What is meant here by explicit expression? Which constants are well known? I remember a competition question that involved line intersections in a complicated curve where I had to ask the professors if we needed to prove the Jordan curve theorem or could assume it. We had not been taught it formally yet. It pays to require clarity. $\endgroup$ – ex0du5 Dec 12 '13 at 21:52
Write $\rho = \frac12 + i \gamma$ for a nontrivial zero of a primitive Lfunction. ("Primitive" means that it can't be written as the product of other Lfunctions.)
It is generally believed that:
a) If $\gamma\not=0$ then $\gamma$ is transcendental.
b) If $\gamma\not=0$ then $\gamma$ is algebraically independent of every wellknown constant and every other zero of every primitive Lfunction (except when the Lfunction has real coefficients, in which case $\frac12  i \gamma$ is also a zero).
As far as I know, nobody has any clue how to prove these conjectures.
Clarification added later: what definition of Lfunction are we using?
Greg Martin's comment (below) refers to $L(s+ i y)$ where $L(s)$ is an Lfunction and $y$ is real. While it is true that for some definitions of "Lfunction" the set of Lfunctions is closed under that operation, that is not what I intended.
For the Lfunctions in my answer above, the Euler product axiom can be written as:
There is a Dirichlet character $\chi$, the "central character" of the Lfunction, such that \begin{equation} L(s)= \prod_{p \, {\rm prime}} F_p(p^{s})^{1}, \end{equation} where $F_p$ is a polynomial of the form \begin{equation} F_p(z)=1a_p z + \cdots + (1)^d\chi(p) z^d . \end{equation}
Here $d$ is the degree of the Lfunction. Note that I have normalized the Lfunction so that the functional equation relates $s$ to $1s$.
All known Lfunctions satisfy that axiom, and this formulation tells you how to select the distinguished member of the family $L(s+i y)$.

$\begingroup$ Clarification question: does "primitive" exclude, or does it include, nonabelian Artin Lfunctions with ArtinBrauer factorizations into ratios of products? Or, e.g., is this contingent on unproven things about their actual holomorphy? $\endgroup$ – paul garrett Mar 18 '13 at 18:25

$\begingroup$ At at a colloquium I once heard the comment, "Can we even prove that all the zeros of every primitive Lfunction are not all rational multiples of each other?" $\endgroup$ – Eric Naslund Mar 18 '13 at 18:33

1$\begingroup$ Indeed, the comment Eric alludes to was made (more or less) by Lior Silberman. One has to be careful, though, since $L(s+ic)$ is also a primitive $L$function for any real $c$ whenever $L$ is a primitive $L$function without a pole at $s=1$. So one needs to make more modest formulations that don't fall to this loophole: the set of Dirichlet $L$functions, for example. $\endgroup$ – Greg Martin Mar 18 '13 at 18:54

$\begingroup$ If an Artin Lfunction was genuinely a ratio, and not a product, of other Lfunctions (in other words, if it had infinitely many poles) then it does not deserve to be called an Lfunction! For this discussion I think it makes sense to assume all reasonable conjectures. $\endgroup$ – David Farmer Mar 18 '13 at 19:06

$\begingroup$ I'm totally happy to assume all reasonable conjectures! :) But what I mean is, whether it is presumed (or known?) that an express for an Artin Lfunction as ratio of product of abelian Lfunctions, which has no poles at all (say), has no denominator at all (in a reasonable sense)? That is, that the lack of poles will never be due to cancellation of zeros of numerator and denominator? $\endgroup$ – paul garrett Mar 18 '13 at 19:15
There is no explicit value for the imaginary part of the nth zero. However it satisfies a simple transcendental equation for each n, whose solution is well approximated by the Lambert function. See LeClair and Franca on arXiv, math.NT