| L(s) = 1 | − 88.8·2-s + 243·3-s + 5.84e3·4-s + 1.13e4·5-s − 2.15e4·6-s + 2.30e4·7-s − 3.36e5·8-s + 5.90e4·9-s − 1.01e6·10-s − 2.40e5·11-s + 1.41e6·12-s − 2.32e6·13-s − 2.04e6·14-s + 2.76e6·15-s + 1.79e7·16-s + 1.04e7·17-s − 5.24e6·18-s − 1.05e7·19-s + 6.65e7·20-s + 5.59e6·21-s + 2.13e7·22-s − 3.94e7·23-s − 8.18e7·24-s + 8.10e7·25-s + 2.06e8·26-s + 1.43e7·27-s + 1.34e8·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.85·4-s + 1.63·5-s − 1.13·6-s + 0.517·7-s − 3.63·8-s + 0.333·9-s − 3.20·10-s − 0.450·11-s + 1.64·12-s − 1.73·13-s − 1.01·14-s + 0.941·15-s + 4.28·16-s + 1.78·17-s − 0.654·18-s − 0.973·19-s + 4.65·20-s + 0.298·21-s + 0.883·22-s − 1.27·23-s − 2.09·24-s + 1.65·25-s + 3.41·26-s + 0.192·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 + 88.8T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.13e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.30e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 2.40e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.32e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.04e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.05e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.94e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.37e6T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.38e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.44e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.07e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 9.52e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.34e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.54e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 5.48e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.63e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.43e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.08e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.30e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.78e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.44e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.26e11T + 7.15e21T^{2} \) |
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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
−9.971271525466922830246406280125, −9.398870717911985039232258387556, −8.221257538257625884164023650511, −7.57307857531527330321756891372, −6.40390973150463781731167166651, −5.34728346316276497165828430406, −2.84441592446991006132670738105, −2.08340185152388931214214548302, −1.42463785645027513281915268896, 0,
1.42463785645027513281915268896, 2.08340185152388931214214548302, 2.84441592446991006132670738105, 5.34728346316276497165828430406, 6.40390973150463781731167166651, 7.57307857531527330321756891372, 8.221257538257625884164023650511, 9.398870717911985039232258387556, 9.971271525466922830246406280125
