| L(s) = 1 | + 0.227·2-s + 8.23·3-s − 7.94·4-s + 2.75·5-s + 1.87·6-s − 21.5·7-s − 3.63·8-s + 40.8·9-s + 0.626·10-s − 54.5·11-s − 65.4·12-s − 52.4·13-s − 4.90·14-s + 22.6·15-s + 62.7·16-s + 9.31·18-s − 19.6·19-s − 21.8·20-s − 177.·21-s − 12.4·22-s − 13.9·23-s − 29.9·24-s − 117.·25-s − 11.9·26-s + 114.·27-s + 171.·28-s − 70.0·29-s + ⋯ |
| L(s) = 1 | + 0.0805·2-s + 1.58·3-s − 0.993·4-s + 0.246·5-s + 0.127·6-s − 1.16·7-s − 0.160·8-s + 1.51·9-s + 0.0198·10-s − 1.49·11-s − 1.57·12-s − 1.11·13-s − 0.0936·14-s + 0.390·15-s + 0.980·16-s + 0.121·18-s − 0.237·19-s − 0.244·20-s − 1.84·21-s − 0.120·22-s − 0.126·23-s − 0.254·24-s − 0.939·25-s − 0.0902·26-s + 0.815·27-s + 1.15·28-s − 0.448·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 0.227T + 8T^{2} \) |
| 3 | \( 1 - 8.23T + 27T^{2} \) |
| 5 | \( 1 - 2.75T + 125T^{2} \) |
| 7 | \( 1 + 21.5T + 343T^{2} \) |
| 11 | \( 1 + 54.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 19.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 70.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 207.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 576.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 78.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 213.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 29.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 855.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 13.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 651.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−10.31061198015101676946120161048, −9.771599860831218389070432067517, −9.085867252535256590985416747097, −8.103470613941702047583939995062, −7.36166463822212989203209777305, −5.74169250777063201584768053264, −4.45150824708205712483916662329, −3.26309476058344674878165990297, −2.40543195498166811251681261515, 0,
2.40543195498166811251681261515, 3.26309476058344674878165990297, 4.45150824708205712483916662329, 5.74169250777063201584768053264, 7.36166463822212989203209777305, 8.103470613941702047583939995062, 9.085867252535256590985416747097, 9.771599860831218389070432067517, 10.31061198015101676946120161048
