| L(s) = 1 | − 33.5·5-s + 67.1·7-s + 121·11-s + 504.·13-s − 984.·17-s + 281.·19-s − 359.·23-s − 1.99e3·25-s + 5.02e3·29-s − 7.01e3·31-s − 2.25e3·35-s − 5.24e3·37-s + 1.38e4·41-s + 2.01e4·43-s − 6.78e3·47-s − 1.22e4·49-s + 2.72e4·53-s − 4.05e3·55-s − 1.90e4·59-s + 2.40e4·61-s − 1.69e4·65-s + 5.32e4·67-s + 4.42e4·71-s + 2.19e4·73-s + 8.12e3·77-s − 2.63e4·79-s + 1.94e4·83-s + ⋯ |
| L(s) = 1 | − 0.600·5-s + 0.518·7-s + 0.301·11-s + 0.828·13-s − 0.826·17-s + 0.178·19-s − 0.141·23-s − 0.639·25-s + 1.10·29-s − 1.31·31-s − 0.310·35-s − 0.629·37-s + 1.28·41-s + 1.66·43-s − 0.447·47-s − 0.731·49-s + 1.33·53-s − 0.180·55-s − 0.714·59-s + 0.827·61-s − 0.496·65-s + 1.44·67-s + 1.04·71-s + 0.481·73-s + 0.156·77-s − 0.475·79-s + 0.309·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.934588370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934588370\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 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
−10.73237771126989262526609621461, −9.444563667083650407927268892953, −8.575805838877039743851838944940, −7.76969573023446593815273655828, −6.75840766658514869916072316675, −5.67643691692985461560033926788, −4.44864318704261173955482499609, −3.60357999334401235192699182660, −2.09907183802404112960617002281, −0.74625772058557356819582919855,
0.74625772058557356819582919855, 2.09907183802404112960617002281, 3.60357999334401235192699182660, 4.44864318704261173955482499609, 5.67643691692985461560033926788, 6.75840766658514869916072316675, 7.76969573023446593815273655828, 8.575805838877039743851838944940, 9.444563667083650407927268892953, 10.73237771126989262526609621461
