| L(s) = 1 | + 2.20·2-s + 16.8·3-s − 27.1·4-s + 75.2·5-s + 37.1·6-s − 225.·7-s − 130.·8-s + 40.5·9-s + 166.·10-s + 121·11-s − 456.·12-s + 455.·13-s − 498.·14-s + 1.26e3·15-s + 579.·16-s + 190.·17-s + 89.5·18-s − 135.·19-s − 2.04e3·20-s − 3.79e3·21-s + 267.·22-s + 2.79e3·23-s − 2.19e3·24-s + 2.53e3·25-s + 1.00e3·26-s − 3.40e3·27-s + 6.11e3·28-s + ⋯ |
| L(s) = 1 | + 0.390·2-s + 1.08·3-s − 0.847·4-s + 1.34·5-s + 0.421·6-s − 1.73·7-s − 0.721·8-s + 0.166·9-s + 0.525·10-s + 0.301·11-s − 0.915·12-s + 0.747·13-s − 0.679·14-s + 1.45·15-s + 0.565·16-s + 0.160·17-s + 0.0651·18-s − 0.0860·19-s − 1.14·20-s − 1.87·21-s + 0.117·22-s + 1.10·23-s − 0.779·24-s + 0.810·25-s + 0.291·26-s − 0.899·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.645807385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645807385\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 2.20T + 32T^{2} \) |
| 3 | \( 1 - 16.8T + 243T^{2} \) |
| 5 | \( 1 - 75.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 225.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 455.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 190.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 135.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.46e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.31e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.30e4T + 8.58e9T^{2} \) |
| 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
−19.54182348511164404261224713935, −18.37338410671450228598153863535, −16.78119483762825834620605882746, −14.83242559757902066159747415911, −13.52417057545944201388601644620, −13.07491618127576319982311433708, −9.746999007057604474252975238105, −8.993526732987570460136666348473, −6.07389680191523228400090157786, −3.23172583586239099777043617004,
3.23172583586239099777043617004, 6.07389680191523228400090157786, 8.993526732987570460136666348473, 9.746999007057604474252975238105, 13.07491618127576319982311433708, 13.52417057545944201388601644620, 14.83242559757902066159747415911, 16.78119483762825834620605882746, 18.37338410671450228598153863535, 19.54182348511164404261224713935
