When the index 
is real, the functions 
, 
, 
, and 
each have an infinite number of real zeros, all of
which are simple with the possible exception of 
. For nonnegative 
, the 
th positive zeros of these functions are denoted 
, 
, 
, and 
, respectively, except that 
is typically counted as the first zero of 
(Abramowitz and Stegun 1972, p. 370).
The first few roots 
of the Bessel function 
are given in the following table for small nonnegative
integer values of 
and 
.
They can be found in the Wolfram Language
using the command BesselJZero[n,
k].
 |  |  |  |  |  |  |
| 1 | 2.4048 | 3.8317 | 5.1356 | 6.3802 | 7.5883 | 8.7715 |
| 2 | 5.5201 | 7.0156 | 8.4172 | 9.7610 | 11.0647 | 12.3386 |
| 3 | 8.6537 | 10.1735 | 11.6198 | 13.0152 | 14.3725 | 15.7002 |
| 4 | 11.7915 | 13.3237 | 14.7960 | 16.2235 | 17.6160 | 18.9801 |
| 5 | 14.9309 | 16.4706 | 17.9598 | 19.4094 | 20.8269 | 22.2178 |
The first few roots 
of the derivative of the Bessel function 
are given in the following table for small nonnegative
integer values of 
and 
.
Versions of the Wolfram Language prior
to 6 implemented these zeros as BesselJPrimeZeros[n, k] in
the BesselZeros package which is now available for separate download (Wolfram
Research). Note that contrary to Abramowitz and Stegun (1972, p. 370), the Wolfram Language defines the first zero
of 
to be approximately 3.8317 rather than zero.
 |  |  |  |  |  |  |
| 1 | 3.8317 | 1.8412 | 3.0542 | 4.2012 | 5.3175 | 6.4156 |
| 2 | 7.0156 | 5.3314 | 6.7061 | 8.0152 | 9.2824 | 10.5199 |
| 3 | 10.1735 | 8.5363 | 9.9695 | 11.3459 | 12.6819 | 13.9872 |
| 4 | 13.3237 | 11.7060 | 13.1704 | 14.5858 | 15.9641 | 17.3128 |
| 5 | 16.4706 | 14.8636 | 16.3475 | 17.7887 | 19.1960 | 20.5755 |
."
Quart. J. Mech. Appl. Math. 1, 220-224, 1948.Olver, F. W. J.
(Ed.). "Zeros and Associated Values." Royal Society Mathematical Tables,
Vol. 7: Bessel Functions. Cambridge, England: Cambridge University Press,
1960.Wolfram Research. "Wolfram Language & System Documentation
Center: Upgrading from NumericalMath BesselZeros."