| L(s) = 1 | + 8·2-s + 33.7·3-s + 64·4-s − 351.·5-s + 269.·6-s + 343·7-s + 512·8-s − 1.04e3·9-s − 2.81e3·10-s + 761.·11-s + 2.15e3·12-s + 2.19e3·13-s + 2.74e3·14-s − 1.18e4·15-s + 4.09e3·16-s + 2.45e4·17-s − 8.39e3·18-s − 3.12e4·19-s − 2.25e4·20-s + 1.15e4·21-s + 6.09e3·22-s − 9.43e4·23-s + 1.72e4·24-s + 4.56e4·25-s + 1.75e4·26-s − 1.09e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.721·3-s + 0.5·4-s − 1.25·5-s + 0.509·6-s + 0.377·7-s + 0.353·8-s − 0.480·9-s − 0.890·10-s + 0.172·11-s + 0.360·12-s + 0.277·13-s + 0.267·14-s − 0.907·15-s + 0.250·16-s + 1.20·17-s − 0.339·18-s − 1.04·19-s − 0.629·20-s + 0.272·21-s + 0.122·22-s − 1.61·23-s + 0.254·24-s + 0.584·25-s + 0.196·26-s − 1.06·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 3 | \( 1 - 33.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 351.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 761.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.82e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.17e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.01e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.39e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.40e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.41e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.94e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.83e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 9.39e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.80e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.29e6T + 8.07e13T^{2} \) |
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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
−11.20896519331190384080134774287, −9.919929303700430192642906336939, −8.241595603086059044255475009542, −8.112013360934595606626934560125, −6.68646329342309694659004859733, −5.31891611198480726380991500319, −3.97081811479384076144051226984, −3.33491075063667943812289319916, −1.86533226246139937776432309410, 0,
1.86533226246139937776432309410, 3.33491075063667943812289319916, 3.97081811479384076144051226984, 5.31891611198480726380991500319, 6.68646329342309694659004859733, 8.112013360934595606626934560125, 8.241595603086059044255475009542, 9.919929303700430192642906336939, 11.20896519331190384080134774287
