| L(s) = 1 | + 16.8·2-s + 154.·4-s + 125·5-s − 286.·7-s + 448.·8-s + 2.10e3·10-s − 6.07e3·11-s + 8.43e3·13-s − 4.82e3·14-s − 1.22e4·16-s + 3.52e4·17-s − 2.71e4·19-s + 1.93e4·20-s − 1.02e5·22-s − 6.43e4·23-s + 1.56e4·25-s + 1.41e5·26-s − 4.43e4·28-s − 4.01e4·29-s + 360.·31-s − 2.63e5·32-s + 5.93e5·34-s − 3.58e4·35-s − 2.94e5·37-s − 4.57e5·38-s + 5.60e4·40-s + 3.23e5·41-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 1.20·4-s + 0.447·5-s − 0.316·7-s + 0.309·8-s + 0.664·10-s − 1.37·11-s + 1.06·13-s − 0.469·14-s − 0.748·16-s + 1.74·17-s − 0.909·19-s + 0.540·20-s − 2.04·22-s − 1.10·23-s + 0.199·25-s + 1.58·26-s − 0.381·28-s − 0.305·29-s + 0.00217·31-s − 1.42·32-s + 2.58·34-s − 0.141·35-s − 0.956·37-s − 1.35·38-s + 0.138·40-s + 0.732·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 2 | \( 1 - 16.8T + 128T^{2} \) |
| 7 | \( 1 + 286.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.43e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.52e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.71e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.01e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 360.T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.94e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.88e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.03e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.70e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.18e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.12e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.05e7T + 8.07e13T^{2} \) |
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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
−9.907860340310882156947610464613, −8.586920734161535365127025855889, −7.58357315426713186089801060722, −6.25815542027379856556661918026, −5.74748187245070818130373471219, −4.85475458722358499684592339237, −3.67113116036971980257029657240, −2.91047146291227721455433302026, −1.71845477677017337899129573180, 0,
1.71845477677017337899129573180, 2.91047146291227721455433302026, 3.67113116036971980257029657240, 4.85475458722358499684592339237, 5.74748187245070818130373471219, 6.25815542027379856556661918026, 7.58357315426713186089801060722, 8.586920734161535365127025855889, 9.907860340310882156947610464613
