| L(s) = 1 | + 165.·2-s + 729·3-s + 1.92e4·4-s − 4.53e4·5-s + 1.20e5·6-s + 5.05e5·7-s + 1.83e6·8-s + 5.31e5·9-s − 7.52e6·10-s − 4.65e6·11-s + 1.40e7·12-s − 3.25e7·13-s + 8.37e7·14-s − 3.30e7·15-s + 1.46e8·16-s − 8.81e7·17-s + 8.80e7·18-s − 2.09e8·19-s − 8.75e8·20-s + 3.68e8·21-s − 7.72e8·22-s − 1.11e9·23-s + 1.34e9·24-s + 8.39e8·25-s − 5.39e9·26-s + 3.87e8·27-s + 9.74e9·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.35·4-s − 1.29·5-s + 1.05·6-s + 1.62·7-s + 2.48·8-s + 0.333·9-s − 2.37·10-s − 0.792·11-s + 1.35·12-s − 1.86·13-s + 2.97·14-s − 0.750·15-s + 2.18·16-s − 0.886·17-s + 0.610·18-s − 1.02·19-s − 3.05·20-s + 0.937·21-s − 1.45·22-s − 1.57·23-s + 1.43·24-s + 0.687·25-s − 3.42·26-s + 0.192·27-s + 3.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 - 165.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 4.53e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 5.05e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 4.65e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.25e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 8.81e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.09e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.11e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.51e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.88e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 4.42e8T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.71e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.64e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 6.94e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.06e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 6.28e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.93e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.59e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.32e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.93e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 5.25e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.71e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 7.81e12T + 6.73e25T^{2} \) |
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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
−10.31582493130136406842491187348, −8.219768093516591033449970548827, −7.76743265690420860665381277101, −6.78516015622285592079326776450, −5.14262804483160542012805152445, −4.57015150338992377263450056879, −3.91071280566062324347050834687, −2.53059515952652912761465903549, −1.98591232812066276563785643596, 0,
1.98591232812066276563785643596, 2.53059515952652912761465903549, 3.91071280566062324347050834687, 4.57015150338992377263450056879, 5.14262804483160542012805152445, 6.78516015622285592079326776450, 7.76743265690420860665381277101, 8.219768093516591033449970548827, 10.31582493130136406842491187348
