| L(s) = 1 | − 10.0·2-s − 21.3·3-s + 68.7·4-s − 25·5-s + 214.·6-s + 45.5·7-s − 368.·8-s + 211.·9-s + 250.·10-s + 121·11-s − 1.46e3·12-s + 590.·13-s − 457.·14-s + 533.·15-s + 1.50e3·16-s + 545.·17-s − 2.12e3·18-s + 361·19-s − 1.71e3·20-s − 972.·21-s − 1.21e3·22-s + 253.·23-s + 7.86e3·24-s + 625·25-s − 5.92e3·26-s + 661.·27-s + 3.13e3·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.36·3-s + 2.14·4-s − 0.447·5-s + 2.42·6-s + 0.351·7-s − 2.03·8-s + 0.872·9-s + 0.793·10-s + 0.301·11-s − 2.93·12-s + 0.969·13-s − 0.623·14-s + 0.611·15-s + 1.46·16-s + 0.457·17-s − 1.54·18-s + 0.229·19-s − 0.960·20-s − 0.481·21-s − 0.534·22-s + 0.0998·23-s + 2.78·24-s + 0.200·25-s − 1.71·26-s + 0.174·27-s + 0.755·28-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 | \( 1 + 10.0T + 32T^{2} \) |
| 3 | \( 1 + 21.3T + 243T^{2} \) |
| 7 | \( 1 - 45.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 590.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 545.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 253.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.51e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.64e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.80e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.14e4T + 8.58e9T^{2} \) |
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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
−8.678445530768700199336222901684, −8.130221511482251525029171587703, −7.16263847466732852856846382310, −6.48481661688032668333238809805, −5.70168200253842510919576327297, −4.56645567383541044648663057823, −3.15083545387500045242845179440, −1.58709230476431045011816314576, −0.905062124032014144931665398231, 0,
0.905062124032014144931665398231, 1.58709230476431045011816314576, 3.15083545387500045242845179440, 4.56645567383541044648663057823, 5.70168200253842510919576327297, 6.48481661688032668333238809805, 7.16263847466732852856846382310, 8.130221511482251525029171587703, 8.678445530768700199336222901684
