| L(s) = 1 | + 19.2·3-s + 33.5·5-s + 67.1·7-s + 128.·9-s − 121·11-s + 504.·13-s + 646.·15-s + 984.·17-s + 281.·19-s + 1.29e3·21-s + 359.·23-s − 1.99e3·25-s − 2.20e3·27-s − 5.02e3·29-s − 7.01e3·31-s − 2.33e3·33-s + 2.25e3·35-s − 5.24e3·37-s + 9.72e3·39-s − 1.38e4·41-s + 2.01e4·43-s + 4.30e3·45-s + 6.78e3·47-s − 1.22e4·49-s + 1.89e4·51-s − 2.72e4·53-s − 4.05e3·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s + 0.600·5-s + 0.518·7-s + 0.528·9-s − 0.301·11-s + 0.828·13-s + 0.741·15-s + 0.826·17-s + 0.178·19-s + 0.640·21-s + 0.141·23-s − 0.639·25-s − 0.583·27-s − 1.10·29-s − 1.31·31-s − 0.372·33-s + 0.310·35-s − 0.629·37-s + 1.02·39-s − 1.28·41-s + 1.66·43-s + 0.316·45-s + 0.447·47-s − 0.731·49-s + 1.02·51-s − 1.33·53-s − 0.180·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.612515601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.612515601\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 19.2T + 243T^{2} \) |
| 5 | \( 1 - 33.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 67.1T + 1.68e4T^{2} \) |
| 13 | \( 1 - 504.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 984.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 281.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 359.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−14.61651902086917308250277065826, −13.91003979504429233601366491875, −12.90304618011021512932371739624, −11.20296495813208287730505807015, −9.758382979192415083282011700810, −8.671280379876782484776180263602, −7.56968891098292746947632903673, −5.60741289320492949595702011850, −3.53367842901175626558992789064, −1.87064333848924947222738007843,
1.87064333848924947222738007843, 3.53367842901175626558992789064, 5.60741289320492949595702011850, 7.56968891098292746947632903673, 8.671280379876782484776180263602, 9.758382979192415083282011700810, 11.20296495813208287730505807015, 12.90304618011021512932371739624, 13.91003979504429233601366491875, 14.61651902086917308250277065826
