| L(s) = 1 | + 48.6·2-s + 45.3·3-s + 317.·4-s + 3.25e3·5-s + 2.20e3·6-s + 1.37e4·7-s − 8.41e4·8-s − 1.75e5·9-s + 1.58e5·10-s − 6.46e5·11-s + 1.44e4·12-s + 9.19e5·13-s + 6.69e5·14-s + 1.47e5·15-s − 4.74e6·16-s − 1.92e6·17-s − 8.51e6·18-s + 7.77e6·19-s + 1.03e6·20-s + 6.24e5·21-s − 3.14e7·22-s − 3.87e7·23-s − 3.82e6·24-s − 3.82e7·25-s + 4.47e7·26-s − 1.59e7·27-s + 4.37e6·28-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.107·3-s + 0.155·4-s + 0.465·5-s + 0.115·6-s + 0.309·7-s − 0.908·8-s − 0.988·9-s + 0.500·10-s − 1.20·11-s + 0.0167·12-s + 0.686·13-s + 0.332·14-s + 0.0502·15-s − 1.13·16-s − 0.328·17-s − 1.06·18-s + 0.720·19-s + 0.0722·20-s + 0.0333·21-s − 1.30·22-s − 1.25·23-s − 0.0979·24-s − 0.783·25-s + 0.738·26-s − 0.214·27-s + 0.0480·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 - 1.15e8T \) |
| good | 2 | \( 1 - 48.6T + 2.04e3T^{2} \) |
| 3 | \( 1 - 45.3T + 1.77e5T^{2} \) |
| 5 | \( 1 - 3.25e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.37e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.46e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 9.19e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.92e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.77e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.87e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.87e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.31e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.47e7T + 1.77e17T^{2} \) |
| 43 | \( 1 - 1.49e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 6.72e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.93e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.37e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 4.56e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 7.37e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 8.44e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.12e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.69e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.16e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.02e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.17e11T + 7.15e21T^{2} \) |
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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
−13.40579408816529844400192070540, −12.10725448783031598136154132158, −10.90070842853413409119312140624, −9.302114407932229087554507561824, −7.996823735319181462004391781650, −5.99619979085455917689951504859, −5.22172450055622483162626247592, −3.62547776774777899327183807429, −2.28357979451254450017461283206, 0,
2.28357979451254450017461283206, 3.62547776774777899327183807429, 5.22172450055622483162626247592, 5.99619979085455917689951504859, 7.996823735319181462004391781650, 9.302114407932229087554507561824, 10.90070842853413409119312140624, 12.10725448783031598136154132158, 13.40579408816529844400192070540
