| L(s) = 1 | + 0.290·2-s + 9·3-s − 31.9·4-s + 87.1·5-s + 2.61·6-s + 167.·7-s − 18.5·8-s + 81·9-s + 25.2·10-s + 252.·11-s − 287.·12-s − 289.·13-s + 48.6·14-s + 783.·15-s + 1.01e3·16-s − 227.·17-s + 23.4·18-s − 1.16e3·19-s − 2.78e3·20-s + 1.50e3·21-s + 73.3·22-s − 1.23e3·23-s − 166.·24-s + 4.46e3·25-s − 83.9·26-s + 729·27-s − 5.34e3·28-s + ⋯ |
| L(s) = 1 | + 0.0512·2-s + 0.577·3-s − 0.997·4-s + 1.55·5-s + 0.0296·6-s + 1.29·7-s − 0.102·8-s + 0.333·9-s + 0.0799·10-s + 0.629·11-s − 0.575·12-s − 0.474·13-s + 0.0662·14-s + 0.899·15-s + 0.992·16-s − 0.190·17-s + 0.0170·18-s − 0.738·19-s − 1.55·20-s + 0.746·21-s + 0.0323·22-s − 0.485·23-s − 0.0591·24-s + 1.42·25-s − 0.0243·26-s + 0.192·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.112503158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.112503158\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 0.290T + 32T^{2} \) |
| 5 | \( 1 - 87.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 167.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 252.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 289.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 227.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.73e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 998.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.09e5T + 8.58e9T^{2} \) |
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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
−11.94957383148589325517568654644, −10.47050920996282805680145543981, −9.681465861749473721689752122374, −8.831332089985402727810564841659, −8.010804132379282906951652328551, −6.38407365787007087625432191740, −5.17028757549476092791269257381, −4.26189064364207279541971671456, −2.38463552422994502808857675954, −1.23781975604350924910722443042,
1.23781975604350924910722443042, 2.38463552422994502808857675954, 4.26189064364207279541971671456, 5.17028757549476092791269257381, 6.38407365787007087625432191740, 8.010804132379282906951652328551, 8.831332089985402727810564841659, 9.681465861749473721689752122374, 10.47050920996282805680145543981, 11.94957383148589325517568654644
