| L(s) = 1 | − 9.17·2-s − 14.3·3-s + 52.1·4-s + 25·5-s + 131.·6-s − 184.·8-s − 37.7·9-s − 229.·10-s + 215.·11-s − 746.·12-s − 132.·13-s − 358.·15-s + 25.7·16-s − 441.·17-s + 346.·18-s − 519.·19-s + 1.30e3·20-s − 1.97e3·22-s − 4.38e3·23-s + 2.64e3·24-s + 625·25-s + 1.21e3·26-s + 4.02e3·27-s + 7.11e3·29-s + 3.28e3·30-s + 1.03e4·31-s + 5.67e3·32-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.918·3-s + 1.62·4-s + 0.447·5-s + 1.49·6-s − 1.02·8-s − 0.155·9-s − 0.725·10-s + 0.536·11-s − 1.49·12-s − 0.217·13-s − 0.410·15-s + 0.0251·16-s − 0.370·17-s + 0.252·18-s − 0.329·19-s + 0.728·20-s − 0.870·22-s − 1.73·23-s + 0.937·24-s + 0.200·25-s + 0.353·26-s + 1.06·27-s + 1.57·29-s + 0.666·30-s + 1.92·31-s + 0.979·32-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 | \( 1 + 9.17T + 32T^{2} \) |
| 3 | \( 1 + 14.3T + 243T^{2} \) |
| 11 | \( 1 - 215.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 132.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 441.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 519.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 184.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.27e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.36e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 343.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.93e4T + 8.58e9T^{2} \) |
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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
−10.35823664440669116004405138490, −10.01936183388751312296585562969, −8.796451265190235964710497529490, −8.073827851770022363720358759396, −6.67523632518757145198837294599, −6.12851511555963442465164727059, −4.60858058437535111832877427032, −2.48386794152010896139907774190, −1.13127032162188693646222583250, 0,
1.13127032162188693646222583250, 2.48386794152010896139907774190, 4.60858058437535111832877427032, 6.12851511555963442465164727059, 6.67523632518757145198837294599, 8.073827851770022363720358759396, 8.796451265190235964710497529490, 10.01936183388751312296585562969, 10.35823664440669116004405138490
