The xi-function is the function
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the Riemann zeta function and 
is the gamma function
(Hardy 1999, p. 41; Gradshteyn and Ryzhik 2000, p. 1076; Edwards 2001,
p. 16). This is a variant of the function originally
defined by Riemann in his landmark paper (Riemann 1859), where the above now standard
notation follows Landau (Edwards 2001, p. 16).
The apparent singularities in the defining expressions are filled in by analytic continuation, giving an entire function (Edwards 2001, p. 16).
It is implemented in the Wolfram Language as RiemannXi[s].
The zeros of 
and of its derivatives
are all located on the critical strip 
, where 
. Therefore, the nontrivial
zeros of the Riemann zeta function exactly
correspond to those of 
(i.e., the roots of 
are the same as those of 
for real 
), with the additional benefit that 
is purely real.
The first few positive values of 
for which 
are summarized in the following table (Wagon 1991,
pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding
negative values are also roots.
The integers closest to these values are 14, 21, 25,
30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410).
The numbers of zeros 
with 
, 
, 
, ..., counted with multiplicity,
are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080;
Odlyzko).
 | OEIS |  |
| 1 | A058303 | 14.134725 |
| 2 | 21.022040 | |
| 3 | 25.010858 | |
| 4 | 30.424876 | |
| 5 | 32.935062 | |
| 6 | 37.586178 |
Special values include
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The 
function satisfies the functional
equation
 |
(9)
|
(Edwards 2001, p. 16).
The xi-function has the Taylor series about 1/2 of
 |
(10)
|
where
![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) |
(11)
|
and
 |  |  |
(12)
|
 |  | ![1/2[theta_3(0,e^(-pix))-1]](/images/equations/Xi-Function/Inline49.svg) |
(13)
|
(Edwards 2001, p. 15), with 
a Jacobi theta
function. The coefficient 
has the simple analytic form
 |  |  |
(14)
|
 |  |  |
(15)
|
(OEIS A114720).
Liu and Mao (2026) reported that the coefficient sequence 
is infinitely
logarithmically concave. Every entry remains strictly positive after any number
of iterations of 
,
with 
at each iteration. The result does not assume the Riemann
hypothesis. Liu and Mao (2026) report generative AI assistance with approaches,
technical checks, and exposition and state that they verified the proofs and computations.
As stated by Riemann (1859) and first rigorously proved by Hadamard (1893), the xi-function can be written as
 |
(16)
|
where the infinite product runs over the roots 
of 
, counted with multiplicity,
in increasing order of 
,
including each pair of complex conjugates together
(Edwards 2001, pp. 17-21).
The xi-function extended into the complex plane is illustrated above.
The function 
is related to
 |
(17)
|
where 
(Gradshteyn and Ryzhik 2000, p. 1074; Edwards 2001, p. 16), which is the
function originally considered and actually denoted
by Riemann (Edwards 2001, p. 16).
This function can also be defined as
 |
(18)
|
giving
 |
(19)
|
The de Bruijn-Newman constant is defined in terms of the 
function.
Hardy (1914) proved that 
has infinitely many real roots (Hardy's theorem), Hardy
and Littlewood (1921) proved that the number of real roots
between 0 and 
is at least 
for some positive constant 
and all sufficiently large 
, and Selberg (1942) proved that this number is in fact at
least 
for some positive 
and all large 
(Edwards 2001, p. 19).
Coffey (2004) gives a number of formulas of derivatives of 
.
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.Liu,
Y. and Mao, J. "Infinite Log-Concavity of the Taylor Coefficients of the Riemann
xi-Function." 4 Oct 2026. 
th
Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.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 
?" Math. Intel. 8, 57-62, 1986.Wagon,
S.