| L(s) = 1 | − 0.164·2-s − 31.9·4-s + 46.4·5-s + 0.553·7-s + 10.5·8-s − 7.63·10-s − 524.·11-s + 933.·13-s − 0.0908·14-s + 1.02e3·16-s − 462.·17-s − 452.·19-s − 1.48e3·20-s + 86.1·22-s − 216.·23-s − 962.·25-s − 153.·26-s − 17.6·28-s + 841·29-s + 5.16e3·31-s − 504.·32-s + 76.0·34-s + 25.7·35-s + 9.19e3·37-s + 74.3·38-s + 488.·40-s − 1.72e4·41-s + ⋯ |
| L(s) = 1 | − 0.0290·2-s − 0.999·4-s + 0.831·5-s + 0.00426·7-s + 0.0580·8-s − 0.0241·10-s − 1.30·11-s + 1.53·13-s − 0.000123·14-s + 0.997·16-s − 0.388·17-s − 0.287·19-s − 0.831·20-s + 0.0379·22-s − 0.0855·23-s − 0.308·25-s − 0.0444·26-s − 0.00426·28-s + 0.185·29-s + 0.965·31-s − 0.0870·32-s + 0.0112·34-s + 0.00355·35-s + 1.10·37-s + 0.00835·38-s + 0.0482·40-s − 1.60·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - 841T \) |
| good | 2 | \( 1 + 0.164T + 32T^{2} \) |
| 5 | \( 1 - 46.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 0.553T + 1.68e4T^{2} \) |
| 11 | \( 1 + 524.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 933.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 462.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 452.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 216.T + 6.43e6T^{2} \) |
| 31 | \( 1 - 5.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.17e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 291.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 926.T + 8.58e9T^{2} \) |
| 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
−10.38862047150226779273867816773, −9.797969443286499581570604447826, −8.637430916340264804318561875470, −8.053118973323984925989294933678, −6.40070059253316010154432672957, −5.50691160138135576185037708801, −4.47844603985895449937057803810, −3.10344125692618307082168315447, −1.52584575959843792444607327245, 0,
1.52584575959843792444607327245, 3.10344125692618307082168315447, 4.47844603985895449937057803810, 5.50691160138135576185037708801, 6.40070059253316010154432672957, 8.053118973323984925989294933678, 8.637430916340264804318561875470, 9.797969443286499581570604447826, 10.38862047150226779273867816773
