| L(s) = 1 | − 2.30e7·2-s + 3.91e14·4-s + 1.15e16·5-s + 1.54e19·7-s − 5.79e21·8-s − 2.65e23·10-s + 3.19e24·11-s + 1.57e25·13-s − 3.55e26·14-s + 7.86e28·16-s + 3.42e28·17-s + 1.00e30·19-s + 4.51e30·20-s − 7.37e31·22-s − 5.17e31·23-s − 5.78e32·25-s − 3.64e32·26-s + 6.04e33·28-s + 6.90e33·29-s − 1.79e35·31-s − 9.98e35·32-s − 7.90e35·34-s + 1.77e35·35-s − 2.68e36·37-s − 2.30e37·38-s − 6.67e37·40-s − 1.17e38·41-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.78·4-s + 0.431·5-s + 0.212·7-s − 3.47·8-s − 0.839·10-s + 1.07·11-s + 0.104·13-s − 0.414·14-s + 3.97·16-s + 0.416·17-s + 0.889·19-s + 1.20·20-s − 2.09·22-s − 0.517·23-s − 0.813·25-s − 0.203·26-s + 0.593·28-s + 0.297·29-s − 1.61·31-s − 4.25·32-s − 0.809·34-s + 0.0919·35-s − 0.377·37-s − 1.73·38-s − 1.49·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{49}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 2.30e7T + 1.40e14T^{2} \) |
| 5 | \( 1 - 1.15e16T + 7.10e32T^{2} \) |
| 7 | \( 1 - 1.54e19T + 5.24e39T^{2} \) |
| 11 | \( 1 - 3.19e24T + 8.81e48T^{2} \) |
| 13 | \( 1 - 1.57e25T + 2.26e52T^{2} \) |
| 17 | \( 1 - 3.42e28T + 6.77e57T^{2} \) |
| 19 | \( 1 - 1.00e30T + 1.26e60T^{2} \) |
| 23 | \( 1 + 5.17e31T + 1.00e64T^{2} \) |
| 29 | \( 1 - 6.90e33T + 5.40e68T^{2} \) |
| 31 | \( 1 + 1.79e35T + 1.24e70T^{2} \) |
| 37 | \( 1 + 2.68e36T + 5.07e73T^{2} \) |
| 41 | \( 1 + 1.17e38T + 6.32e75T^{2} \) |
| 43 | \( 1 - 1.58e38T + 5.92e76T^{2} \) |
| 47 | \( 1 - 1.34e39T + 3.87e78T^{2} \) |
| 53 | \( 1 + 4.68e40T + 1.09e81T^{2} \) |
| 59 | \( 1 + 3.84e41T + 1.69e83T^{2} \) |
| 61 | \( 1 - 4.37e41T + 8.13e83T^{2} \) |
| 67 | \( 1 - 4.64e42T + 6.69e85T^{2} \) |
| 71 | \( 1 + 1.91e43T + 1.02e87T^{2} \) |
| 73 | \( 1 - 8.25e43T + 3.76e87T^{2} \) |
| 79 | \( 1 + 5.45e44T + 1.54e89T^{2} \) |
| 83 | \( 1 - 1.02e45T + 1.57e90T^{2} \) |
| 89 | \( 1 + 1.06e46T + 4.18e91T^{2} \) |
| 97 | \( 1 - 3.72e46T + 2.38e93T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14809578249569981391468017186, −9.897800751517235379924629585085, −9.153901196737174295705014449731, −7.976501082254238841568430056123, −6.88953555679555684518578691550, −5.76357935859784668155267069387, −3.38684999671004069184653407645, −1.92292327156562728193686883674, −1.24444822465184357253644947432, 0,
1.24444822465184357253644947432, 1.92292327156562728193686883674, 3.38684999671004069184653407645, 5.76357935859784668155267069387, 6.88953555679555684518578691550, 7.976501082254238841568430056123, 9.153901196737174295705014449731, 9.897800751517235379924629585085, 11.14809578249569981391468017186
