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.Brent, R. P.; 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. II." Math. Comput.39, 681-688, 1982.Coffey, M. W. "Relations and Positivity Results for Derivatives of the Riemann Function." J. Comput. Appl. Math.166, 525-534, 2004.Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc.50, 341-353, 2003. http://www.ams.org/notices/200303/fea-conrey-web.pdf.Edwards, H. M. "The Function ." §1.8 in Riemann's Zeta Function. New York: Dover, pp. 16-18, 2001.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, corr. enl. 4th ed. San Diego, CA: Academic Press, 2000.Hadamard, J. "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann." J. math. pures appl.9, 171-215, 1893.Hardy, G. H. "Sur les zéros de la fonction de Riemann." C. R. Acad. Sci. Paris158, 1012-1014, 1914.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Littlewood, J. E. "The Zeros of Riemann's Zeta-Function on the Critical Line." Math. Z.10, 283-317, 1921.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 202-203, 2003.Keiper, J. B. "Power Series Expansions of Riemann's Function." Math. Comput.58, 765-773, 1992.Li, X.-J. "The Positivity of a Sequence of Numbers and the Riemann Hypothesis." J. Number Th.65, 325-333, 1997.Odlyzko, A. M. "The th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Odlyzko, A. "Tables of Zeros of the Riemann Zeta Function." http://www.dtc.umn.edu/~odlyzko/zeta_tables/.Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972. Also reprinted in English translation in Edwards, H. M. Appendix. Riemann's Zeta Function. New York: Dover, pp. 299-305, 2001.Selberg, A. "On the Zeros of Riemann's Zeta-Function." Skr. Norske Vid.-Akad. Oslo, No. 10, 1942.Sloane, N. J. A. Sequences A002410, A058303, A072080, and A114720 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.Wagon, S. "The Evidence: Where Are the Zeros of Zeta of ?" Math. Intel.8, 57-62, 1986.Wagon, S. Mathematica in Action. New York: W. H. Freeman, 1991.
是 黎曼 zeta 函数，而 
是 伽玛函数 (Gradshteyn and Ryzhik 2000, p. 1076; Hardy 1999, p. 41; Edwards 2001, p. 16)。这是黎曼在其里程碑式的论文 (Riemann 1859) 中最初定义的函数的变体，其中上述现在标准的符号表示法遵循 Landau (Edwards 2001, p. 16)。
的零点及其 导数 的零点都位于 临界带 
上，其中 
。因此，黎曼 zeta 函数 的非平凡零点与 
的零点完全对应 (即，
的根与实数 
的 
的根相同)，另外的好处是 
是纯实数。
、
、... 的零点数量分别为 0、29、649、10142、138069、1747146、... (OEIS A072080; Odlyzko)。
函数满足 函数方程
![a_(2n)=4int_1^infty(d[x^(3/2)psi^'(x)])/(dx)((1/2lnx)^(2n))/((2n)!)x^(-1/4)dx](/images/equations/Xi-Function/NumberedEquation3.svg)
![1/2[theta_3(0,e^(-pix))-1]](/images/equations/Xi-Function/Inline45.svg)
是 雅可比 theta 函数。系数 
具有简单的解析形式
的根 
(Edwards 2001, pp. 17-21)。
与以下项相关
(Gradshteyn and Ryzhik 2000, p. 1074; Edwards 2001, p. 16)，这是黎曼最初考虑并实际表示为 
的函数 (Edwards 2001, p. 16)。此函数也可以定义为
函数定义的。
有无数个实根 (Hardy 定理)，Hardy 和 Littlewood (1921) 证明 0 到 
之间的实根数至少为 
，其中 
为某个正常数，
为所有足够大的值，Selberg (1942) 证明，事实上，这个数字至少为 
，其中 
为某个正数，
为所有大的值 (Edwards 2001, p. 19)。
导数的许多公式。
Function." J. Comput. Appl. Math. 166, 525-534, 2004.Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. 
." §1.8 in 
de Riemann." C. R. Acad. Sci. Paris 158, 1012-1014, 1914.Hardy, G. H. 
Function." Math. Comput. 58, 765-773, 1992.Li, X.-J. "The Positivity of a Sequence of Numbers and the Riemann Hypothesis." J. Number Th. 65, 325-333, 1997.Odlyzko, A. M. "The 
th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Odlyzko, A. "Tables of Zeros of the Riemann Zeta Function." 
?" Math. Intel. 8, 57-62, 1986.Wagon, S.