| L(s) = 1 | + 5.38·2-s + 9·3-s − 3.02·4-s + 53.0·5-s + 48.4·6-s + 107.·7-s − 188.·8-s + 81·9-s + 285.·10-s − 580.·11-s − 27.1·12-s + 15.9·13-s + 577.·14-s + 477.·15-s − 918.·16-s − 1.77e3·17-s + 436.·18-s − 2.16e3·19-s − 160.·20-s + 965.·21-s − 3.12e3·22-s − 3.07e3·23-s − 1.69e3·24-s − 309.·25-s + 85.8·26-s + 729·27-s − 324.·28-s + ⋯ |
| L(s) = 1 | + 0.951·2-s + 0.577·3-s − 0.0944·4-s + 0.949·5-s + 0.549·6-s + 0.827·7-s − 1.04·8-s + 0.333·9-s + 0.903·10-s − 1.44·11-s − 0.0545·12-s + 0.0261·13-s + 0.787·14-s + 0.547·15-s − 0.896·16-s − 1.49·17-s + 0.317·18-s − 1.37·19-s − 0.0896·20-s + 0.477·21-s − 1.37·22-s − 1.21·23-s − 0.601·24-s − 0.0991·25-s + 0.0248·26-s + 0.192·27-s − 0.0781·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 157 | \( 1 - 2.46e4T \) |
| good | 2 | \( 1 - 5.38T + 32T^{2} \) |
| 5 | \( 1 - 53.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 580.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 15.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.77e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.56e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.84e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.63e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.17e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.11e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.50e4T + 8.58e9T^{2} \) |
| 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
−9.893410251357394262868552465151, −8.622112802917193506645128653653, −8.264778515312799066973082699746, −6.73911380523929156972983847561, −5.78697441368077864978975574942, −4.86392518077740486023523268456, −4.14086270564889470873417322608, −2.62829791860656420959413898542, −2.00545127850753001664244749892, 0,
2.00545127850753001664244749892, 2.62829791860656420959413898542, 4.14086270564889470873417322608, 4.86392518077740486023523268456, 5.78697441368077864978975574942, 6.73911380523929156972983847561, 8.264778515312799066973082699746, 8.622112802917193506645128653653, 9.893410251357394262868552465151
