| L(s) = 1 | + 2-s − 15·4-s + 49·7-s − 31·8-s + 81·9-s − 206·11-s + 49·14-s + 209·16-s + 81·18-s − 206·22-s − 734·23-s + 625·25-s − 735·28-s + 1.23e3·29-s + 705·32-s − 1.21e3·36-s − 1.29e3·37-s − 334·43-s + 3.09e3·44-s − 734·46-s + 2.40e3·49-s + 625·50-s − 5.58e3·53-s − 1.51e3·56-s + 1.23e3·58-s + 3.96e3·63-s − 2.63e3·64-s + ⋯ |
| L(s) = 1 | + 1/4·2-s − 0.937·4-s + 7-s − 0.484·8-s + 9-s − 1.70·11-s + 1/4·14-s + 0.816·16-s + 1/4·18-s − 0.425·22-s − 1.38·23-s + 25-s − 0.937·28-s + 1.46·29-s + 0.688·32-s − 0.937·36-s − 0.945·37-s − 0.180·43-s + 1.59·44-s − 0.346·46-s + 49-s + 1/4·50-s − 1.98·53-s − 0.484·56-s + 0.366·58-s + 63-s − 0.644·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9246613506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9246613506\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - T + p^{4} T^{2} \) |
| 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 + 206 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 + 734 T + p^{4} T^{2} \) |
| 29 | \( 1 - 1234 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 1294 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 334 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 + 5582 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 - 4946 T + p^{4} T^{2} \) |
| 71 | \( 1 - 2914 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 + 3646 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−21.81883397137813991055040757330, −20.83385690170942964759789219046, −18.62050119086863269536489308862, −17.81169047302870193601462621196, −15.66739993002832371047384600194, −14.02308025680361225275427813264, −12.62044328517825495351964360153, −10.29318726257942305906272690701, −8.111867345315333483666297511038, −4.86176241637417627999224132563,
4.86176241637417627999224132563, 8.111867345315333483666297511038, 10.29318726257942305906272690701, 12.62044328517825495351964360153, 14.02308025680361225275427813264, 15.66739993002832371047384600194, 17.81169047302870193601462621196, 18.62050119086863269536489308862, 20.83385690170942964759789219046, 21.81883397137813991055040757330
