| L(s) = 1 | + 7.81·2-s + 29.0·4-s + 74.2·5-s − 22.8·8-s + 580.·10-s + 424.·11-s − 252.·13-s − 1.10e3·16-s + 1.10e3·17-s − 6.47·19-s + 2.15e3·20-s + 3.31e3·22-s + 3.61e3·23-s + 2.39e3·25-s − 1.97e3·26-s + 5.00e3·29-s + 2.82e3·31-s − 7.93e3·32-s + 8.63e3·34-s − 2.04e3·37-s − 50.5·38-s − 1.69e3·40-s − 9.39e3·41-s + 1.03e4·43-s + 1.23e4·44-s + 2.82e4·46-s − 1.70e4·47-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.908·4-s + 1.32·5-s − 0.126·8-s + 1.83·10-s + 1.05·11-s − 0.413·13-s − 1.08·16-s + 0.926·17-s − 0.00411·19-s + 1.20·20-s + 1.46·22-s + 1.42·23-s + 0.765·25-s − 0.571·26-s + 1.10·29-s + 0.527·31-s − 1.36·32-s + 1.28·34-s − 0.245·37-s − 0.00568·38-s − 0.167·40-s − 0.872·41-s + 0.851·43-s + 0.960·44-s + 1.96·46-s − 1.12·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.337207481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.337207481\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 7.81T + 32T^{2} \) |
| 5 | \( 1 - 74.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 424.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 6.47T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.38e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−10.31627747553299632771650627440, −9.514489204450618791343020776871, −8.676118986873133704593962268968, −7.02664568108799308881009367171, −6.29710337840667688276505477340, −5.45643236240654797602436610653, −4.68112288608020926837452705748, −3.43911455345558934617982142119, −2.44961379065162586697594535564, −1.16085660965213121240531385096,
1.16085660965213121240531385096, 2.44961379065162586697594535564, 3.43911455345558934617982142119, 4.68112288608020926837452705748, 5.45643236240654797602436610653, 6.29710337840667688276505477340, 7.02664568108799308881009367171, 8.676118986873133704593962268968, 9.514489204450618791343020776871, 10.31627747553299632771650627440
