| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 114.·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s + 913.·10-s − 4.32e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 3.08e3·15-s + 4.09e3·16-s − 5.47e3·17-s − 5.83e3·18-s + 3.10e4·19-s − 7.30e3·20-s + 9.26e3·21-s + 3.45e4·22-s − 4.09e4·23-s − 1.38e4·24-s − 6.50e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.288·10-s − 0.979·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.235·15-s + 0.250·16-s − 0.270·17-s − 0.235·18-s + 1.03·19-s − 0.204·20-s + 0.218·21-s + 0.692·22-s − 0.701·23-s − 0.204·24-s − 0.833·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 5 | \( 1 + 114.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.32e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 5.47e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.61e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.51e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.01e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.98e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.15e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.47e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.10e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.71e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.29e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.71e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.12e7T + 8.07e13T^{2} \) |
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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
−9.149994169664441817474960657306, −8.159203987721388978190939875455, −7.83014473739082016414872905609, −6.80558620820588183462035937689, −5.60047519578141886260287134500, −4.45047958381446750722661393550, −3.25669769505401763524987391694, −2.32034271108852319783765651775, −1.18463145882928958294761094077, 0,
1.18463145882928958294761094077, 2.32034271108852319783765651775, 3.25669769505401763524987391694, 4.45047958381446750722661393550, 5.60047519578141886260287134500, 6.80558620820588183462035937689, 7.83014473739082016414872905609, 8.159203987721388978190939875455, 9.149994169664441817474960657306
