where
is the Rogers L-function and are integers not all equal to 0 (Gordon and Mcintosh 1997).
Loxton (1991, p. 289) gives a slew of similar identities having rational coefficients

(2)

instead of integers.

The only known -algebraic
numbers of order 1 are

(3)

(4)

(5)

(6)

(7)

(Loxton 1991, pp. 287 and 289; Bytsko 1999), where .

The only known rational -algebraic numbers are 1/2 and 1/3:

(8)

(9)

(Lewin 1982, pp. 317-318; Gordon and McIntosh 1997).

There are a number of known quadratic -algebraic numbers. Watson (1937) found

Bytsko, A. G. "Two-Term Dilogarithm Identities Related to Conformal Field Theory." 9 Nov 1999. http://arxiv.org/abs/math-ph/9911012.Gordon,
B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan
J.1, 431-448, 1997.Lewin, L. "The Dilogarithm in Algebraic
Fields." J. Austral. Math. Soc. Ser. A33, 302-330, 1982.Lewin,
L. (Ed.). Structural
Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.Loxton,
J. H. "Special Values of the Dilogarithm Function." Acta Arith.43,
155-166, 1984.Loxton, J. H. "Partition Identities and the
Dilogarithm." Ch. 13 in Structural
Properties of Polylogarithms (Ed. L. Lewin). Providence, RI: Amer. Math.
Soc., pp. 287-299, 1991.Watson, G. N. "A Note on Spence's
Logarithmic Transcendent." Quart. J. Math. Oxford Ser.8, 39-42,
1937.