| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 462.·5-s + 216·6-s + 343·7-s + 512·8-s + 729·9-s − 3.69e3·10-s + 721.·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 1.24e4·15-s + 4.09e3·16-s + 2.07e4·17-s + 5.83e3·18-s − 2.52e4·19-s − 2.95e4·20-s + 9.26e3·21-s + 5.77e3·22-s + 1.13e4·23-s + 1.38e4·24-s + 1.35e5·25-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 − 1.65·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.16·10-s + 0.163·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.955·15-s + 0.250·16-s + 1.02·17-s + 0.235·18-s − 0.843·19-s − 0.827·20-s + 0.218·21-s + 0.115·22-s + 0.193·23-s + 0.204·24-s + 1.73·25-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 | \( 1 + 462.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 721.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.52e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.13e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.96e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.76e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.06e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.83e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.41e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.56e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.74e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.42e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.17e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.78e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.79e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.95e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.37e7T + 8.07e13T^{2} \) |
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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
−9.017572789233800948890843121829, −8.053750150750001956114694258357, −7.58661042973140508862908319078, −6.65576175552575709822157157806, −5.26674390957929513998988647240, −4.28231602135126732879219447223, −3.66868615758597697898631458301, −2.72220301391384497041255264239, −1.35166618176504357570163949549, 0,
1.35166618176504357570163949549, 2.72220301391384497041255264239, 3.66868615758597697898631458301, 4.28231602135126732879219447223, 5.26674390957929513998988647240, 6.65576175552575709822157157806, 7.58661042973140508862908319078, 8.053750150750001956114694258357, 9.017572789233800948890843121829
