| L(s) = 1 | − 9.63·2-s − 23.2·3-s + 60.7·4-s + 58.9·5-s + 224.·6-s + 49·7-s − 276.·8-s + 298.·9-s − 567.·10-s − 1.41e3·12-s + 705.·13-s − 471.·14-s − 1.37e3·15-s + 721.·16-s − 1.71e3·17-s − 2.87e3·18-s − 794.·19-s + 3.58e3·20-s − 1.14e3·21-s − 3.39e3·23-s + 6.44e3·24-s + 350.·25-s − 6.79e3·26-s − 1.28e3·27-s + 2.97e3·28-s − 1.72e3·29-s + 1.32e4·30-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.49·3-s + 1.89·4-s + 1.05·5-s + 2.54·6-s + 0.377·7-s − 1.52·8-s + 1.22·9-s − 1.79·10-s − 2.83·12-s + 1.15·13-s − 0.643·14-s − 1.57·15-s + 0.705·16-s − 1.43·17-s − 2.08·18-s − 0.505·19-s + 2.00·20-s − 0.564·21-s − 1.33·23-s + 2.28·24-s + 0.112·25-s − 1.97·26-s − 0.339·27-s + 0.717·28-s − 0.381·29-s + 2.67·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4091102978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4091102978\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 49T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 9.63T + 32T^{2} \) |
| 3 | \( 1 + 23.2T + 243T^{2} \) |
| 5 | \( 1 - 58.9T + 3.12e3T^{2} \) |
| 13 | \( 1 - 705.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 794.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.37e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.63e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.49e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.16e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.45e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e4T + 8.58e9T^{2} \) |
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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
−9.459534418656353442252633865531, −8.821010405456697785116275513640, −7.86087426830514561012395355154, −6.74866515203808261961654301331, −6.20078187109095435667265043121, −5.52319658159204431071787417908, −4.20436737380242775587850184941, −2.15551840698810953302506128156, −1.53228318511398152092789522517, −0.41701979501673128743476090666,
0.41701979501673128743476090666, 1.53228318511398152092789522517, 2.15551840698810953302506128156, 4.20436737380242775587850184941, 5.52319658159204431071787417908, 6.20078187109095435667265043121, 6.74866515203808261961654301331, 7.86087426830514561012395355154, 8.821010405456697785116275513640, 9.459534418656353442252633865531
