| L(s) = 1 | − 27·3-s + 366.·5-s + 355.·7-s + 729·9-s − 7.10e3·11-s + 3.59e3·13-s − 9.90e3·15-s − 2.13e4·17-s + 1.51e4·19-s − 9.58e3·21-s − 9.53e4·23-s + 5.63e4·25-s − 1.96e4·27-s + 1.94e5·29-s − 1.65e4·31-s + 1.91e5·33-s + 1.30e5·35-s − 3.21e5·37-s − 9.71e4·39-s + 1.59e5·41-s + 8.56e5·43-s + 2.67e5·45-s − 7.48e5·47-s − 6.97e5·49-s + 5.75e5·51-s + 3.05e5·53-s − 2.60e6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.31·5-s + 0.391·7-s + 0.333·9-s − 1.60·11-s + 0.454·13-s − 0.757·15-s − 1.05·17-s + 0.506·19-s − 0.225·21-s − 1.63·23-s + 0.721·25-s − 0.192·27-s + 1.47·29-s − 0.0995·31-s + 0.928·33-s + 0.513·35-s − 1.04·37-s − 0.262·39-s + 0.360·41-s + 1.64·43-s + 0.437·45-s − 1.05·47-s − 0.846·49-s + 0.607·51-s + 0.281·53-s − 2.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| good | 5 | \( 1 - 366.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 355.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.10e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.59e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.13e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.51e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.53e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.94e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.65e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.21e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.56e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.48e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.05e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.88e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.25e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.43e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.56e7T + 8.07e13T^{2} \) |
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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
−10.52928043253335344299832105621, −10.09261273768634625519763126852, −8.804858774849473182387272941986, −7.66224334059995851659278257446, −6.29960472513332279358358937564, −5.56358033686033007544496644508, −4.56207958852901442730845078202, −2.64388946919992546141170401693, −1.58872695378975618013156499611, 0,
1.58872695378975618013156499611, 2.64388946919992546141170401693, 4.56207958852901442730845078202, 5.56358033686033007544496644508, 6.29960472513332279358358937564, 7.66224334059995851659278257446, 8.804858774849473182387272941986, 10.09261273768634625519763126852, 10.52928043253335344299832105621
