| L(s) = 1 | − 6.41·2-s − 86.8·4-s − 258.·5-s − 1.37e3·7-s + 1.37e3·8-s + 1.65e3·10-s + 628.·11-s − 664.·13-s + 8.81e3·14-s + 2.27e3·16-s + 1.65e4·17-s + 4.01e4·19-s + 2.24e4·20-s − 4.03e3·22-s − 1.21e4·23-s − 1.13e4·25-s + 4.26e3·26-s + 1.19e5·28-s + 1.08e5·29-s + 2.28e5·31-s − 1.91e5·32-s − 1.05e5·34-s + 3.55e5·35-s − 5.75e5·37-s − 2.57e5·38-s − 3.56e5·40-s + 4.25e5·41-s + ⋯ |
| L(s) = 1 | − 0.566·2-s − 0.678·4-s − 0.924·5-s − 1.51·7-s + 0.951·8-s + 0.524·10-s + 0.142·11-s − 0.0839·13-s + 0.859·14-s + 0.139·16-s + 0.814·17-s + 1.34·19-s + 0.627·20-s − 0.0807·22-s − 0.208·23-s − 0.144·25-s + 0.0475·26-s + 1.02·28-s + 0.822·29-s + 1.37·31-s − 1.03·32-s − 0.461·34-s + 1.40·35-s − 1.86·37-s − 0.761·38-s − 0.880·40-s + 0.964·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 6.41T + 128T^{2} \) |
| 5 | \( 1 + 258.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.37e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 628.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 664.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.65e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.01e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.08e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.75e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.25e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.33e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.27e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.69e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.45e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.47e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.74e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.20e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.14e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.99e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.23e7T + 8.07e13T^{2} \) |
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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
−10.18637265712235250134536137808, −9.764300296392214615827830863609, −8.670080820382475031217104798700, −7.74421402076399992225646405779, −6.76363820408350181764688488345, −5.33645745597534049667048324960, −3.97162006428039486377606880742, −3.12522412108677712426975020085, −0.968536617725518669212287500374, 0,
0.968536617725518669212287500374, 3.12522412108677712426975020085, 3.97162006428039486377606880742, 5.33645745597534049667048324960, 6.76363820408350181764688488345, 7.74421402076399992225646405779, 8.670080820382475031217104798700, 9.764300296392214615827830863609, 10.18637265712235250134536137808
