| L(s) = 1 | − 4.98·2-s − 6.26·3-s + 16.8·4-s − 15.7·5-s + 31.1·6-s − 0.789·7-s − 43.8·8-s + 12.2·9-s + 78.4·10-s − 45.3·11-s − 105.·12-s + 46.7·13-s + 3.93·14-s + 98.6·15-s + 84.1·16-s − 60.8·18-s + 100.·19-s − 264.·20-s + 4.94·21-s + 225.·22-s + 84.1·23-s + 274.·24-s + 122.·25-s − 232.·26-s + 92.5·27-s − 13.2·28-s + 101.·29-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.20·3-s + 2.10·4-s − 1.40·5-s + 2.12·6-s − 0.0426·7-s − 1.93·8-s + 0.452·9-s + 2.48·10-s − 1.24·11-s − 2.53·12-s + 0.997·13-s + 0.0751·14-s + 1.69·15-s + 1.31·16-s − 0.796·18-s + 1.21·19-s − 2.95·20-s + 0.0514·21-s + 2.18·22-s + 0.762·23-s + 2.33·24-s + 0.983·25-s − 1.75·26-s + 0.659·27-s − 0.0896·28-s + 0.651·29-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 | \( 1 + 4.98T + 8T^{2} \) |
| 3 | \( 1 + 6.26T + 27T^{2} \) |
| 5 | \( 1 + 15.7T + 125T^{2} \) |
| 7 | \( 1 + 0.789T + 343T^{2} \) |
| 11 | \( 1 + 45.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.36T + 2.97e4T^{2} \) |
| 37 | \( 1 - 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 260.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 398.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 126.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 150.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 434.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 493.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 72.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 711.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3T + 9.12e5T^{2} \) |
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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
−10.84288836304842221689307581694, −10.19563978306596774451278292762, −8.855993962389258887347445653460, −8.005638007745934770164711981566, −7.32450235182404201690501034731, −6.25624765704336276733321327982, −4.95073924827432450003647869775, −3.11927256052460660064871427172, −0.977322334493618334195441898523, 0,
0.977322334493618334195441898523, 3.11927256052460660064871427172, 4.95073924827432450003647869775, 6.25624765704336276733321327982, 7.32450235182404201690501034731, 8.005638007745934770164711981566, 8.855993962389258887347445653460, 10.19563978306596774451278292762, 10.84288836304842221689307581694
