| L(s) = 1 | − 7.22·2-s − 15.9·3-s + 20.2·4-s − 25·5-s + 115.·6-s + 85.0·8-s + 10.4·9-s + 180.·10-s − 387.·11-s − 322.·12-s − 920.·13-s + 398.·15-s − 1.26e3·16-s + 678.·17-s − 75.7·18-s + 2.76e3·19-s − 505.·20-s + 2.79e3·22-s + 2.06e3·23-s − 1.35e3·24-s + 625·25-s + 6.65e3·26-s + 3.70e3·27-s − 2.92e3·29-s − 2.87e3·30-s + 4.25e3·31-s + 6.40e3·32-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 1.02·3-s + 0.632·4-s − 0.447·5-s + 1.30·6-s + 0.469·8-s + 0.0431·9-s + 0.571·10-s − 0.964·11-s − 0.645·12-s − 1.51·13-s + 0.456·15-s − 1.23·16-s + 0.569·17-s − 0.0550·18-s + 1.75·19-s − 0.282·20-s + 1.23·22-s + 0.814·23-s − 0.479·24-s + 0.200·25-s + 1.93·26-s + 0.977·27-s − 0.645·29-s − 0.583·30-s + 0.795·31-s + 1.10·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 7.22T + 32T^{2} \) |
| 3 | \( 1 + 15.9T + 243T^{2} \) |
| 11 | \( 1 + 387.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 920.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 678.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.76e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 154.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.47e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.00e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.07e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.66e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.70e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.54922614687735474133115350162, −9.927446489189857789710214167547, −8.888711862044523054936002580983, −7.63242626740884179345743680725, −7.23301421550558077010934328792, −5.55496298657086659619928415561, −4.76439962533707293214923117743, −2.77393816521697384147327131406, −0.936751823763627328388798543442, 0,
0.936751823763627328388798543442, 2.77393816521697384147327131406, 4.76439962533707293214923117743, 5.55496298657086659619928415561, 7.23301421550558077010934328792, 7.63242626740884179345743680725, 8.888711862044523054936002580983, 9.927446489189857789710214167547, 10.54922614687735474133115350162
