Special Values
Some special values include:
where G represents Catalan's constant, and
where in the above is an example of the polygamma function. More generally, for any positive integer k:
where represent the Euler numbers. For integer k ≥ 0, this extends to:
Hence, the function vanishes for all odd negative integral values of the argument.
| s | approximate value β(s) | OEIS |
|---|---|---|
| 1/5 | 0.5737108471859466493572665 | |
| 1/4 | 0.5907230564424947318659591 | |
| 1/3 | 0.6178550888488520660725389 | |
| 1/2 | 0.6676914571896091766586909 | A195103 |
| 1 | 0.7853981633974483096156608 | A003881 |
| 2 | 0.9159655941772190150546035 | A006752 |
| 3 | 0.9689461462593693804836348 | A153071 |
| 4 | 0.9889445517411053361084226 | A175572 |
| 5 | 0.9961578280770880640063194 | A175571 |
| 6 | 0.9986852222184381354416008 | A175570 |
| 7 | 0.9995545078905399094963465 | |
| 8 | 0.9998499902468296563380671 | |
| 9 | 0.9999496841872200898213589 | |
| 10 | 0.9999831640261968774055407 |
Read more about this topic: Dirichlet Beta Function
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