| L(s) = 1 | + 77.8·2-s − 243·3-s + 4.01e3·4-s + 5.39e3·5-s − 1.89e4·6-s − 4.28e4·7-s + 1.53e5·8-s + 5.90e4·9-s + 4.20e5·10-s − 1.04e5·11-s − 9.75e5·12-s − 1.47e6·13-s − 3.33e6·14-s − 1.31e6·15-s + 3.71e6·16-s + 8.81e6·17-s + 4.59e6·18-s + 6.43e6·19-s + 2.16e7·20-s + 1.04e7·21-s − 8.14e6·22-s − 4.37e7·23-s − 3.72e7·24-s − 1.97e7·25-s − 1.14e8·26-s − 1.43e7·27-s − 1.71e8·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.96·4-s + 0.772·5-s − 0.993·6-s − 0.962·7-s + 1.65·8-s + 0.333·9-s + 1.32·10-s − 0.195·11-s − 1.13·12-s − 1.09·13-s − 1.65·14-s − 0.445·15-s + 0.884·16-s + 1.50·17-s + 0.573·18-s + 0.596·19-s + 1.51·20-s + 0.555·21-s − 0.337·22-s − 1.41·23-s − 0.954·24-s − 0.403·25-s − 1.89·26-s − 0.192·27-s − 1.88·28-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 | \( 1 - 77.8T + 2.04e3T^{2} \) |
| 5 | \( 1 - 5.39e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.28e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.04e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.47e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 8.81e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.43e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.37e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.52e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.63e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.81e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.35e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 3.37e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 5.89e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.20e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 6.83e8T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.58e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 8.59e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.85e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.26e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.45e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.58e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.65e10T + 7.15e21T^{2} \) |
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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
−10.21930658105727865752987285519, −9.682864017160521852787101245015, −7.58558515111424858054046248679, −6.56707572615739794729983040058, −5.71028239542102278719270305111, −5.15438719399183140310602635364, −3.82088032359173884376059668508, −2.86460326360819418600341123272, −1.73176009269148747249902002999, 0,
1.73176009269148747249902002999, 2.86460326360819418600341123272, 3.82088032359173884376059668508, 5.15438719399183140310602635364, 5.71028239542102278719270305111, 6.56707572615739794729983040058, 7.58558515111424858054046248679, 9.682864017160521852787101245015, 10.21930658105727865752987285519
