| L(s) = 1 | − 5.27·2-s + 3·3-s + 19.8·4-s + 10.5·5-s − 15.8·6-s + 7·7-s − 62.3·8-s + 9·9-s − 55.6·10-s + 34.7·11-s + 59.4·12-s − 37.2·13-s − 36.9·14-s + 31.6·15-s + 170.·16-s − 10.5·17-s − 47.4·18-s − 58.5·19-s + 209.·20-s + 21·21-s − 183.·22-s − 125.·23-s − 187.·24-s − 13.7·25-s + 196.·26-s + 27·27-s + 138.·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.47·4-s + 0.943·5-s − 1.07·6-s + 0.377·7-s − 2.75·8-s + 0.333·9-s − 1.75·10-s + 0.952·11-s + 1.43·12-s − 0.795·13-s − 0.704·14-s + 0.544·15-s + 2.66·16-s − 0.150·17-s − 0.621·18-s − 0.707·19-s + 2.33·20-s + 0.218·21-s − 1.77·22-s − 1.13·23-s − 1.59·24-s − 0.109·25-s + 1.48·26-s + 0.192·27-s + 0.936·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6948668260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6948668260\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 7T \) |
| good | 2 | \( 1 + 5.27T + 8T^{2} \) |
| 5 | \( 1 - 10.5T + 125T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 - 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
| 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
−17.57422945386022424369784705375, −17.04459096373884350284302281883, −15.48248129646803441342299119708, −14.11173331179018455937931545626, −11.95403214843741524545736968147, −10.29660072704947938309086752171, −9.382639145802876843343958417977, −8.177937000406163814747537789084, −6.57897767162756355612751119591, −1.97603540068635921626971898304,
1.97603540068635921626971898304, 6.57897767162756355612751119591, 8.177937000406163814747537789084, 9.382639145802876843343958417977, 10.29660072704947938309086752171, 11.95403214843741524545736968147, 14.11173331179018455937931545626, 15.48248129646803441342299119708, 17.04459096373884350284302281883, 17.57422945386022424369784705375
