| L(s) = 1 | − 7.57·2-s + 20.2·3-s + 25.3·4-s − 153.·6-s − 57.9·7-s + 50.6·8-s + 165.·9-s − 590.·11-s + 511.·12-s + 810.·13-s + 438.·14-s − 1.19e3·16-s − 289·17-s − 1.25e3·18-s + 270.·19-s − 1.17e3·21-s + 4.47e3·22-s + 4.20e3·23-s + 1.02e3·24-s − 6.13e3·26-s − 1.56e3·27-s − 1.46e3·28-s − 404.·29-s + 2.00e3·31-s + 7.41e3·32-s − 1.19e4·33-s + 2.18e3·34-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 1.29·3-s + 0.790·4-s − 1.73·6-s − 0.446·7-s + 0.279·8-s + 0.681·9-s − 1.47·11-s + 1.02·12-s + 1.32·13-s + 0.597·14-s − 1.16·16-s − 0.242·17-s − 0.912·18-s + 0.171·19-s − 0.579·21-s + 1.97·22-s + 1.65·23-s + 0.362·24-s − 1.77·26-s − 0.412·27-s − 0.353·28-s − 0.0893·29-s + 0.373·31-s + 1.27·32-s − 1.90·33-s + 0.324·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 + 289T \) |
| good | 2 | \( 1 + 7.57T + 32T^{2} \) |
| 3 | \( 1 - 20.2T + 243T^{2} \) |
| 7 | \( 1 + 57.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 590.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 810.T + 3.71e5T^{2} \) |
| 19 | \( 1 - 270.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 404.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.08e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.91e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.98e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.00e5T + 8.58e9T^{2} \) |
| 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
−9.613936190180570402601493677584, −8.921976477436901209044235040722, −8.287732483273354550246632455975, −7.63989104354629167809648535226, −6.60224761114251707583277773667, −5.04739134081240491787755185442, −3.49814968206217302594647812402, −2.59701991416362787736119853663, −1.37997443670429947244890021258, 0,
1.37997443670429947244890021258, 2.59701991416362787736119853663, 3.49814968206217302594647812402, 5.04739134081240491787755185442, 6.60224761114251707583277773667, 7.63989104354629167809648535226, 8.287732483273354550246632455975, 8.921976477436901209044235040722, 9.613936190180570402601493677584
