| L(s) = 1 | + 156.·2-s + 729·3-s + 1.61e4·4-s − 5.21e4·5-s + 1.13e5·6-s − 1.21e5·7-s + 1.24e6·8-s + 5.31e5·9-s − 8.14e6·10-s − 5.92e6·11-s + 1.17e7·12-s + 3.41e7·13-s − 1.90e7·14-s − 3.80e7·15-s + 6.16e7·16-s + 1.67e8·17-s + 8.29e7·18-s − 1.20e8·19-s − 8.43e8·20-s − 8.88e7·21-s − 9.24e8·22-s − 8.89e8·23-s + 9.06e8·24-s + 1.50e9·25-s + 5.33e9·26-s + 3.87e8·27-s − 1.97e9·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.577·3-s + 1.97·4-s − 1.49·5-s + 0.995·6-s − 0.391·7-s + 1.67·8-s + 0.333·9-s − 2.57·10-s − 1.00·11-s + 1.13·12-s + 1.96·13-s − 0.675·14-s − 0.862·15-s + 0.918·16-s + 1.68·17-s + 0.574·18-s − 0.588·19-s − 2.94·20-s − 0.226·21-s − 1.73·22-s − 1.25·23-s + 0.968·24-s + 1.23·25-s + 3.38·26-s + 0.192·27-s − 0.772·28-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 | \( 1 - 156.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 5.21e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.21e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.92e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.41e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.67e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.20e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 8.89e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.81e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.89e8T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.67e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.57e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.31e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 5.93e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.17e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 6.03e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.22e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.34e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.83e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 5.33e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.78e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.18e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 5.89e12T + 6.73e25T^{2} \) |
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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
−10.20746009571667957986222030086, −8.358280342543167449911593776701, −7.79342659127237530924792237570, −6.56982583591856122596526785696, −5.56128051678031212740512499267, −4.34743842313661866232210298391, −3.47731302737099157325282459153, −3.19723435160871447160884742532, −1.61497462338893873569331882441, 0,
1.61497462338893873569331882441, 3.19723435160871447160884742532, 3.47731302737099157325282459153, 4.34743842313661866232210298391, 5.56128051678031212740512499267, 6.56982583591856122596526785696, 7.79342659127237530924792237570, 8.358280342543167449911593776701, 10.20746009571667957986222030086
