On evaluation formulas for double L-values

Abstract

In this paper, we give some evaluation formulas for the values of double L-series of Tornheim's type, in terms of the Dirichlet L-values and the Riemann zeta values at positive integers. As special cases, these give the formulas for double L-values given by Terhune.

Double Lerch series and their functional relations

“…This function has been already defined and continued analytically in [16] as a double Dirichlet series (see also [11]). Tsumura in [15] has shown functional relations for L(s 1 , s 2 , s 3 ; χ, ϕ) only when the modulus of the characters are 3 or 4, but in this paper we obtain a functional relation for arbitrary characters.…”

Functional relations and special values of Mordell-Tornheim triple zeta and 𝐿-functions

“…The results in this paper can be regarded as triple analogues of the known results for double zeta and L-functions proved in the papers quoted above and [9,10,13,14,15], and also be regarded as χ-analogues of the results in [5]. Therefore this paper is a step toward the general theory of functional relations for Mordell-Tornheim r-ple zeta and L-functions including known results of r-ple zeta values given in [16].…”

“…Hence, by using (−1) k cos(kπ/2) = i k + i −k /2 (k ∈ Z), we can write (4.6) in the case of θ = π/2 as − L MT,3 (1, 1, 1, 2; 1, 1, 1, χ 4 ) Thus we obtain a new evaluation formula By using the partial fraction decomposition, we are able to see that L MT,3 (1, 1, 1, 2; 1, 1, 1, χ 4 ) coincides with 6 1≤l<m<n χ 4 (n)/lmn 3 , which is an ordinary triple L-value (see [2]). This formula (4.7) can be regarded as a χ-analogue of that of ζ MT,3 (see, for example, [5, Example 3.2]) and as a triple analogue of that of double L-values (see [9,10,15]). Furthermore, by combining (3.12) and (4.7), we can also give an evaluation formula for L MT,3 (1, 1, 2, 1 ; 1, 1, χ 4 , 1) in terms of Dirichlet L-values, double zeta and L-values.…”

On some functional relations between Mordell–Tornheim double L-functions and Dirichlet L-functions

“…Furthermore multiple L-functions have been studied by, for example, Goncharov [8], Akiyama and Ishikawa [1], and Matsumoto and Tanigawa [15]. Recently Terhune evaluated L * MT,2 (p, 0, q; χ, ϕ) (p, q ∈ N) by means of single series under a certain condition (see [18], see also [21]). …”