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It is known that the geometric side of the Selberg trace formula on GL(2) is related to values of quadratic L-functions (due to Sarnak, Zagier, etc).

Are there any conjectures or results about its generalization to other groups, such as GL(3) and GL(n)?

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    $\begingroup$ Do you mean the point that volumes of various compact geodesics are often equal to values of L-functions at $s=1$? Langlands' old work on the residue at the first pole of minimal-parabolic Eisenstein series relates that to the natural volume of the arithmetic quotient, for example. And by this year many other "periods" of Eisenstein series are understood as giving L-functions (although this is not the typical case, in some sense). $\endgroup$
    – paul garrett
    Jan 14, 2017 at 23:49
  • $\begingroup$ to paul garrett Yes, that's what I mean. $\endgroup$
    – 7-adic
    Jan 15, 2017 at 1:13
  • $\begingroup$ to paul garrett the situation for the residue of Eisenstein series is quite different from that of geometric side of trace formulas. I am looking for the connection between geometric side of trace formula on GL(n) and certain special values of certain L-functions. $\endgroup$
    – 7-adic
    Jan 15, 2017 at 2:02

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If I understand correctly what you are looking for, then yes, a fair amount of work has been done. Deitmar and Hoffman use a simple trace formula on SL(3) to get asymptotics of class number of cubic orders, which can also be interpreted as a prime geodesic theorem. This is a higher-dimensional analogue of quadratic class numbers (which can be viewed as special $L$-values) appearing in a trace formula for SL(2). Deitmar also treats higher rank analogues in other papers.

Another way in which zeta/$L$-functions arise in the geometric side of trace formulas (maybe not what you are looking for, but perhaps still interesting) is discussed in general in this paper of Hoffman and for Sp(4) in his joint paper with Wakatsuki. Namely, one gets Shintani zeta functions arising as coefficients in the unipotent geometric terms of the Arthur-Selberg trace formula.

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