| L(s) = 1 | − 31·2-s + 705·4-s + 2.40e3·7-s − 1.39e4·8-s + 6.56e3·9-s + 1.31e4·11-s − 7.44e4·14-s + 2.51e5·16-s − 2.03e5·18-s − 4.07e5·22-s − 2.09e4·23-s + 3.90e5·25-s + 1.69e6·28-s + 1.08e5·29-s − 4.21e6·32-s + 4.62e6·36-s − 2.07e6·37-s − 6.72e6·43-s + 9.27e6·44-s + 6.48e5·46-s + 5.76e6·49-s − 1.21e7·50-s + 1.53e7·53-s − 3.34e7·56-s − 3.35e6·58-s + 1.57e7·63-s + 6.65e7·64-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.75·4-s + 7-s − 3.39·8-s + 9-s + 0.898·11-s − 1.93·14-s + 3.83·16-s − 1.93·18-s − 1.74·22-s − 0.0747·23-s + 25-s + 2.75·28-s + 0.152·29-s − 4.02·32-s + 2.75·36-s − 1.10·37-s − 1.96·43-s + 2.47·44-s + 0.144·46-s + 49-s − 1.93·50-s + 1.94·53-s − 3.39·56-s − 0.296·58-s + 63-s + 3.96·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7050764436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7050764436\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 + 31 T + p^{8} T^{2} \) |
| 3 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 11 | \( 1 - 13154 T + p^{8} T^{2} \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 + 20926 T + p^{8} T^{2} \) |
| 29 | \( 1 - 108194 T + p^{8} T^{2} \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( 1 + 2073886 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 + 6726046 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( 1 - 15377762 T + p^{8} T^{2} \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 + 15839326 T + p^{8} T^{2} \) |
| 71 | \( 1 + 42331966 T + p^{8} T^{2} \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( 1 + 64606846 T + p^{8} T^{2} \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
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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
−20.19919249791923172639004889380, −18.83057030527403870855414681003, −17.78984690345478177936540549681, −16.56263945714005997042348548484, −15.04192493919052488382018448312, −11.80658495895418354406187242632, −10.31503820493995659736389591099, −8.692640034198523192534860102525, −7.06661744652306668936091890853, −1.43975694701657115285982645837,
1.43975694701657115285982645837, 7.06661744652306668936091890853, 8.692640034198523192534860102525, 10.31503820493995659736389591099, 11.80658495895418354406187242632, 15.04192493919052488382018448312, 16.56263945714005997042348548484, 17.78984690345478177936540549681, 18.83057030527403870855414681003, 20.19919249791923172639004889380
