| L(s) = 1 | − 2.70·2-s + 3·3-s − 0.701·4-s − 19.3·5-s − 8.10·6-s − 7·7-s + 23.5·8-s + 9·9-s + 52.2·10-s − 16.2·11-s − 2.10·12-s + 70.0·13-s + 18.9·14-s − 58.0·15-s − 57.8·16-s + 24.6·17-s − 24.3·18-s + 127.·19-s + 13.5·20-s − 21·21-s + 43.9·22-s − 23·23-s + 70.5·24-s + 249.·25-s − 189.·26-s + 27·27-s + 4.91·28-s + ⋯ |
| L(s) = 1 | − 0.955·2-s + 0.577·3-s − 0.0876·4-s − 1.73·5-s − 0.551·6-s − 0.377·7-s + 1.03·8-s + 0.333·9-s + 1.65·10-s − 0.445·11-s − 0.0506·12-s + 1.49·13-s + 0.361·14-s − 0.999·15-s − 0.904·16-s + 0.351·17-s − 0.318·18-s + 1.53·19-s + 0.151·20-s − 0.218·21-s + 0.425·22-s − 0.208·23-s + 0.599·24-s + 1.99·25-s − 1.42·26-s + 0.192·27-s + 0.0331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.70T + 8T^{2} \) |
| 5 | \( 1 + 19.3T + 125T^{2} \) |
| 11 | \( 1 + 16.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 98.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 317.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 390.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 163.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 58.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 25.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 398.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 929.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 303.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 20.9T + 9.12e5T^{2} \) |
| 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
−10.02562859824924931168261933103, −8.936152234808468189986560074037, −8.410007074287913536928053643161, −7.65523948166021950572783036533, −7.01010355813711154498826316161, −5.23274762230016573436888978801, −3.90506151367073209296134336369, −3.32908642876575004106064498578, −1.24932549400161210994993597944, 0,
1.24932549400161210994993597944, 3.32908642876575004106064498578, 3.90506151367073209296134336369, 5.23274762230016573436888978801, 7.01010355813711154498826316161, 7.65523948166021950572783036533, 8.410007074287913536928053643161, 8.936152234808468189986560074037, 10.02562859824924931168261933103
