| L(s) = 1 | + 3·3-s − 16.7·5-s − 7·7-s + 9·9-s + 22.7·11-s + 11.4·13-s − 50.1·15-s + 3.29·17-s + 62.8·19-s − 21·21-s − 76.1·23-s + 154.·25-s + 27·27-s + 216.·29-s − 149.·31-s + 68.1·33-s + 116.·35-s − 187.·37-s + 34.2·39-s − 23.2·41-s − 214.·43-s − 150.·45-s + 187.·47-s + 49·49-s + 9.88·51-s + 122.·53-s − 379.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.49·5-s − 0.377·7-s + 0.333·9-s + 0.622·11-s + 0.243·13-s − 0.862·15-s + 0.0470·17-s + 0.758·19-s − 0.218·21-s − 0.690·23-s + 1.23·25-s + 0.192·27-s + 1.38·29-s − 0.866·31-s + 0.359·33-s + 0.564·35-s − 0.834·37-s + 0.140·39-s − 0.0887·41-s − 0.759·43-s − 0.498·45-s + 0.583·47-s + 0.142·49-s + 0.0271·51-s + 0.317·53-s − 0.929·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 16.7T + 125T^{2} \) |
| 11 | \( 1 - 22.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.29T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 187.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 23.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 353.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 465.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 532.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 185.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
| 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
−8.639967128334873677608722134978, −8.140672270415479573195030832459, −7.25353167953266232328208266408, −6.66959409656147213258350491555, −5.39055774701440019312318478864, −4.19882400508894500242291480939, −3.69231612974098368205611929163, −2.80131738344867543861765690903, −1.27477251061760421743995918458, 0,
1.27477251061760421743995918458, 2.80131738344867543861765690903, 3.69231612974098368205611929163, 4.19882400508894500242291480939, 5.39055774701440019312318478864, 6.66959409656147213258350491555, 7.25353167953266232328208266408, 8.140672270415479573195030832459, 8.639967128334873677608722134978
