| L(s) = 1 | + 1.69·2-s + 12.5·3-s − 29.1·4-s − 25·5-s + 21.2·6-s + 107.·7-s − 103.·8-s − 85.6·9-s − 42.4·10-s + 121·11-s − 365.·12-s + 351.·13-s + 182.·14-s − 313.·15-s + 755.·16-s − 1.40e3·17-s − 145.·18-s + 361·19-s + 728.·20-s + 1.35e3·21-s + 205.·22-s + 1.20e3·23-s − 1.30e3·24-s + 625·25-s + 595.·26-s − 4.12e3·27-s − 3.13e3·28-s + ⋯ |
| L(s) = 1 | + 0.299·2-s + 0.804·3-s − 0.910·4-s − 0.447·5-s + 0.241·6-s + 0.831·7-s − 0.572·8-s − 0.352·9-s − 0.134·10-s + 0.301·11-s − 0.732·12-s + 0.576·13-s + 0.249·14-s − 0.359·15-s + 0.738·16-s − 1.17·17-s − 0.105·18-s + 0.229·19-s + 0.406·20-s + 0.668·21-s + 0.0904·22-s + 0.473·23-s − 0.460·24-s + 0.200·25-s + 0.172·26-s − 1.08·27-s − 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 1.69T + 32T^{2} \) |
| 3 | \( 1 - 12.5T + 243T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 351.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.40e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 644.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e5T + 8.58e9T^{2} \) |
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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
−8.490634338869602205826863808350, −8.412677641887635519777635170288, −7.28026884126584690812533144335, −6.11910896389031392559837134813, −5.08864236443121059893976363650, −4.30201567289716382379123256592, −3.54190484456476123604715881164, −2.54429647260903251036557023495, −1.23505271473526665760611129401, 0,
1.23505271473526665760611129401, 2.54429647260903251036557023495, 3.54190484456476123604715881164, 4.30201567289716382379123256592, 5.08864236443121059893976363650, 6.11910896389031392559837134813, 7.28026884126584690812533144335, 8.412677641887635519777635170288, 8.490634338869602205826863808350
