Berry, M. V. "The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders." Proc. Roy. Soc. London A450, 439-462, 1995.Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." J. Comput. Appl. Math.121, 247-296, 2000.Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput.33, 1361-1372, 1979.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta-Function. Hawthorn, NY: de Gruyter, 1992.Odlyzko, A. M. "The th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Sloane, N. J. A. Sequences A036282, A114721, A114865, and A114866 in "The On-Line Encyclopedia of Integer Sequences."Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. IV." Math. Comput.46, 667-681, 1986.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 143, 1991.
, Riemann-Siegel 
函数定义为
-函数 (Karatsuba and Voronin 1992, Borwein et al. 1999). 顶部的图叠加了 
(粗线) 在 
之上，其中 
是 Riemann zeta 函数。
, Riemann-Siegel theta 函数 
定义为
![I[lnGamma(1/4+1/2it)]-1/2tlnpi](/images/equations/Riemann-SiegelFunctions/Inline11.svg)
![arg[Gamma(1/4+1/2it)]-1/2tlnpi.](/images/equations/Riemann-SiegelFunctions/Inline14.svg)
在 
(OEIS A114865 和 A114866) 处有局部极值。
的值
, 1, ... 被称为 Gram 点 (Edwards 2001, pp. 125-126)。
关于 0 的级数展开式由下式给出
![1/2[-lnpi+psi(1/4)]t-1/(48)psi_2(1/4)t^3+1/(3840)psi_4(1/4)t^5-1/(645120)psi_6(1/4)t^7](/images/equations/Riemann-SiegelFunctions/Inline25.svg)
![-1/4[2gamma+pi+2ln(8pi)]t+1/(24)[pi^3+28zeta(3)]t^3-[(pi^5)/(96)+(31zeta(5))/(10)]+...](/images/equations/Riemann-SiegelFunctions/Inline28.svg)
由
th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Sloane, N. J. A. Sequences