| L(s) = 1 | − 3·2-s + 5·4-s − 7·7-s − 3·8-s + 9·9-s − 6·11-s + 21·14-s − 11·16-s − 27·18-s + 18·22-s + 18·23-s + 25·25-s − 35·28-s − 54·29-s + 45·32-s + 45·36-s − 38·37-s + 58·43-s − 30·44-s − 54·46-s + 49·49-s − 75·50-s − 6·53-s + 21·56-s + 162·58-s − 63·63-s − 91·64-s + ⋯ |
| L(s) = 1 | − 3/2·2-s + 5/4·4-s − 7-s − 3/8·8-s + 9-s − 0.545·11-s + 3/2·14-s − 0.687·16-s − 3/2·18-s + 9/11·22-s + 0.782·23-s + 25-s − 5/4·28-s − 1.86·29-s + 1.40·32-s + 5/4·36-s − 1.02·37-s + 1.34·43-s − 0.681·44-s − 1.17·46-s + 49-s − 3/2·50-s − 0.113·53-s + 3/8·56-s + 2.79·58-s − 63-s − 1.42·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3329817714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3329817714\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p T \) |
| good | 2 | \( 1 + 3 T + p^{2} T^{2} \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 18 T + p^{2} T^{2} \) |
| 29 | \( 1 + 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 + 118 T + p^{2} T^{2} \) |
| 71 | \( 1 - 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−22.54364926152355667236424777146, −20.72321683128359948995825436530, −19.21391450210761761408870427785, −18.40497630679973574791649581702, −16.79600315524904172443777154084, −15.64810467259429164162308409825, −12.96585968300816620901605936259, −10.58097485411185119630259735576, −9.256810748581498587619513382329, −7.21458918128718444354242474222,
7.21458918128718444354242474222, 9.256810748581498587619513382329, 10.58097485411185119630259735576, 12.96585968300816620901605936259, 15.64810467259429164162308409825, 16.79600315524904172443777154084, 18.40497630679973574791649581702, 19.21391450210761761408870427785, 20.72321683128359948995825436530, 22.54364926152355667236424777146
