| L(s) = 1 | − 2.69·2-s − 24.7·4-s + 87.9·5-s + 153.·8-s − 237.·10-s − 41.6·11-s − 989.·13-s + 377.·16-s + 2.12e3·17-s − 928.·19-s − 2.17e3·20-s + 112.·22-s + 4.39e3·23-s + 4.60e3·25-s + 2.67e3·26-s + 3.89e3·29-s − 1.13e3·31-s − 5.91e3·32-s − 5.72e3·34-s − 1.00e4·37-s + 2.50e3·38-s + 1.34e4·40-s + 131.·41-s − 3.53e3·43-s + 1.02e3·44-s − 1.18e4·46-s − 1.60e4·47-s + ⋯ |
| L(s) = 1 | − 0.477·2-s − 0.772·4-s + 1.57·5-s + 0.845·8-s − 0.750·10-s − 0.103·11-s − 1.62·13-s + 0.368·16-s + 1.78·17-s − 0.590·19-s − 1.21·20-s + 0.0495·22-s + 1.73·23-s + 1.47·25-s + 0.774·26-s + 0.859·29-s − 0.211·31-s − 1.02·32-s − 0.849·34-s − 1.20·37-s + 0.281·38-s + 1.33·40-s + 0.0121·41-s − 0.291·43-s + 0.0801·44-s − 0.826·46-s − 1.06·47-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
| \(L(3)\) |
\(\approx\) |
\(1.795188404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.795188404\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.69T + 32T^{2} \) |
| 5 | \( 1 - 87.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 41.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 989.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 928.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 131.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.53e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.60e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.95e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.09e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.84e4T + 8.58e9T^{2} \) |
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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
−9.931742019196789574019984749546, −9.660454156361134452054496457244, −8.725401120603622021765048428445, −7.67059872609948926133210398934, −6.63373168295963908358517163388, −5.28872125427794220489777988982, −4.95418236268717272861502950677, −3.16497839554416782881770923720, −1.90627235126747004305808647995, −0.77297191967750704792356023493,
0.77297191967750704792356023493, 1.90627235126747004305808647995, 3.16497839554416782881770923720, 4.95418236268717272861502950677, 5.28872125427794220489777988982, 6.63373168295963908358517163388, 7.67059872609948926133210398934, 8.725401120603622021765048428445, 9.660454156361134452054496457244, 9.931742019196789574019984749546
