| L(s) = 1 | − 4·2-s + 10·3-s + 16·4-s + 84·5-s − 40·6-s + 49·7-s − 64·8-s − 143·9-s − 336·10-s − 336·11-s + 160·12-s + 584·13-s − 196·14-s + 840·15-s + 256·16-s − 1.45e3·17-s + 572·18-s + 470·19-s + 1.34e3·20-s + 490·21-s + 1.34e3·22-s − 4.20e3·23-s − 640·24-s + 3.93e3·25-s − 2.33e3·26-s − 3.86e3·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.641·3-s + 1/2·4-s + 1.50·5-s − 0.453·6-s + 0.377·7-s − 0.353·8-s − 0.588·9-s − 1.06·10-s − 0.837·11-s + 0.320·12-s + 0.958·13-s − 0.267·14-s + 0.963·15-s + 1/4·16-s − 1.22·17-s + 0.416·18-s + 0.298·19-s + 0.751·20-s + 0.242·21-s + 0.592·22-s − 1.65·23-s − 0.226·24-s + 1.25·25-s − 0.677·26-s − 1.01·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.298596538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298596538\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 10 T + p^{5} T^{2} \) |
| 5 | \( 1 - 84 T + p^{5} T^{2} \) |
| 11 | \( 1 + 336 T + p^{5} T^{2} \) |
| 13 | \( 1 - 584 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 - 470 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 + 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 - 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 + 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 - 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 - 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 - 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 + 16978 T + p^{5} T^{2} \) |
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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
−18.22409149997367053031732153522, −17.64934176165743027115248287728, −16.03645682248407660920089837923, −14.34092229903959090081539461127, −13.28508242726398949406192968580, −10.98017894869982194784676185345, −9.529584197219974550608690543037, −8.262777881296668124181083957358, −5.97738936838904106726730651498, −2.23850347007247423417292537660,
2.23850347007247423417292537660, 5.97738936838904106726730651498, 8.262777881296668124181083957358, 9.529584197219974550608690543037, 10.98017894869982194784676185345, 13.28508242726398949406192968580, 14.34092229903959090081539461127, 16.03645682248407660920089837923, 17.64934176165743027115248287728, 18.22409149997367053031732153522
