| L(s) = 1 | + 120.·2-s + 729·3-s + 6.32e3·4-s − 6.21e4·5-s + 8.78e4·6-s + 1.17e5·7-s − 2.24e5·8-s + 5.31e5·9-s − 7.49e6·10-s − 8.85e6·11-s + 4.61e6·12-s − 8.39e6·13-s + 1.41e7·14-s − 4.53e7·15-s − 7.89e7·16-s + 1.33e8·17-s + 6.40e7·18-s − 3.46e8·19-s − 3.93e8·20-s + 8.57e7·21-s − 1.06e9·22-s + 5.16e8·23-s − 1.63e8·24-s + 2.64e9·25-s − 1.01e9·26-s + 3.87e8·27-s + 7.44e8·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.772·4-s − 1.77·5-s + 0.768·6-s + 0.377·7-s − 0.302·8-s + 0.333·9-s − 2.36·10-s − 1.50·11-s + 0.445·12-s − 0.482·13-s + 0.503·14-s − 1.02·15-s − 1.17·16-s + 1.34·17-s + 0.443·18-s − 1.68·19-s − 1.37·20-s + 0.218·21-s − 2.00·22-s + 0.727·23-s − 0.174·24-s + 2.16·25-s − 0.642·26-s + 0.192·27-s + 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 729T \) |
| 7 | \( 1 - 1.17e5T \) |
| good | 2 | \( 1 - 120.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 6.21e4T + 1.22e9T^{2} \) |
| 11 | \( 1 + 8.85e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 8.39e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.33e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.46e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.16e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.41e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.56e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 4.20e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.82e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.38e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 6.39e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.66e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.62e10T + 1.04e23T^{2} \) |
| 61 | \( 1 + 9.67e9T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.34e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.05e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 9.67e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.46e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 6.16e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 7.71e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 6.34e12T + 6.73e25T^{2} \) |
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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
−14.65590751716431028639040140083, −12.98989220503368054268139818674, −12.19894098090498121735117302799, −10.80780546020790169533284303884, −8.426708799007276381686177857521, −7.34711936090113565869909731791, −5.09984952624912156157940130492, −3.96114119051193182764480743006, −2.79404467674201695885620491172, 0,
2.79404467674201695885620491172, 3.96114119051193182764480743006, 5.09984952624912156157940130492, 7.34711936090113565869909731791, 8.426708799007276381686177857521, 10.80780546020790169533284303884, 12.19894098090498121735117302799, 12.98989220503368054268139818674, 14.65590751716431028639040140083
