| L(s) = 1 | + 2-s + 2·4-s − 4·5-s + 4·7-s + 5·8-s − 4·10-s + 2·11-s + 2·13-s + 4·14-s + 5·16-s − 6·17-s − 4·19-s − 8·20-s + 2·22-s + 6·23-s + 5·25-s + 2·26-s + 8·28-s − 4·29-s − 3·31-s + 10·32-s − 6·34-s − 16·35-s − 3·37-s − 4·38-s − 20·40-s + 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 1.78·5-s + 1.51·7-s + 1.76·8-s − 1.26·10-s + 0.603·11-s + 0.554·13-s + 1.06·14-s + 5/4·16-s − 1.45·17-s − 0.917·19-s − 1.78·20-s + 0.426·22-s + 1.25·23-s + 25-s + 0.392·26-s + 1.51·28-s − 0.742·29-s − 0.538·31-s + 1.76·32-s − 1.02·34-s − 2.70·35-s − 0.493·37-s − 0.648·38-s − 3.16·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.082459885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.082459885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.66325444120093442589088370226, −12.34746182005289072543014347442, −11.52135417211737529592403696610, −11.40145777942347560492962968641, −11.02764885332401663289194776595, −10.91163410613688514849341903148, −10.14691296986573405702794469107, −9.097171371402099595716051490550, −8.491108875433920823632139901996, −8.300079537147651787454175107691, −7.46298497246863215298517608625, −7.31358002864404554212208615440, −6.77741518484778729195316987935, −5.98028871689103106929561727479, −5.06267473736550086588148582320, −4.52109332697467195722041791113, −4.14387406148145574160586639230, −3.63529057435228304430447279733, −2.36587211598253029337752680266, −1.48624052642395739323500039129,
1.48624052642395739323500039129, 2.36587211598253029337752680266, 3.63529057435228304430447279733, 4.14387406148145574160586639230, 4.52109332697467195722041791113, 5.06267473736550086588148582320, 5.98028871689103106929561727479, 6.77741518484778729195316987935, 7.31358002864404554212208615440, 7.46298497246863215298517608625, 8.300079537147651787454175107691, 8.491108875433920823632139901996, 9.097171371402099595716051490550, 10.14691296986573405702794469107, 10.91163410613688514849341903148, 11.02764885332401663289194776595, 11.40145777942347560492962968641, 11.52135417211737529592403696610, 12.34746182005289072543014347442, 12.66325444120093442589088370226
