Let 
be the Riemann-Siegel
function. The unique value 
such that
 |
(1)
|
where 
, 1, ... is then known as a Gram point
(Edwards 2001, pp. 125-126).
An excellent approximation for Gram point 
can be obtained by using the first few terms in the asymptotic
expansion for 
and inverting to obtain
![g_n approx 2piexp[1+W((8n+1)/(8e))],](/images/equations/GramPoint/NumberedEquation2.svg) |
(2)
|
where 
is the Lambert
W-function. This approximation gives as error of 
for 
, decreasing to 
by 
.
The following table gives the first few Gram points.
 | OEIS |  |
| 0 | A114857 | 17.8455995404 |
| 1 | A114858 | 23.1702827012 |
| 2 | 27.6701822178 | |
| 3 | 31.7179799547 | |
| 4 | 35.4671842971 | |
| 5 | 38.9992099640 | |
| 6 | 42.3635503920 | |
| 7 | 45.5930289815 | |
| 8 | 48.7107766217 | |
| 9 | 51.7338428133 | |
| 10 | 54.6752374468 |
The integers closest to these points are 18, 23, 28, 32, 35, 39, 42, 46, 49, 52, 55, 58, ... (OEIS A002505).
There is a unique point at which 
, given by the solution to the equation
 |
(3)
|
and having numerical value
 |
(4)
|
(OEIS A114893).
It is usually the case that ![R[zeta(1/2+ig_n)]=(-1)^nZ(g_n)>0](/images/equations/GramPoint/Inline14.svg)
.
Values of 
for which this does not hold are 
, 134, 195, 211, 232, 254, 288, ... (OEIS A114856),
the first two of which were found by Hutchinson (1925).
de Riemann." Acta Math. 27, 289-304, 1903.Haselgrove,
C. B. and Miller, J. C. P. "Tables of the Riemann Zeta Function."
Royal Society Mathematical Tables, Vol. 6. Cambridge, England: Cambridge
University Press, p. 58, 1960.Hutchinson, J. I. "On the
Roots of the Riemann Zeta-Function." Trans. Amer. Math. Soc. 27,
49-60, 1925.Sloane, N. J. A. Sequences