| L(s) = 1 | + 8.18·2-s − 3.48·3-s + 35.0·4-s − 59.8·5-s − 28.5·6-s + 145.·7-s + 24.8·8-s − 230.·9-s − 490.·10-s + 121·11-s − 122.·12-s + 615.·13-s + 1.18e3·14-s + 208.·15-s − 917.·16-s + 1.84e3·17-s − 1.89e3·18-s + 366.·19-s − 2.09e3·20-s − 505.·21-s + 990.·22-s − 4.51e3·23-s − 86.7·24-s + 459.·25-s + 5.04e3·26-s + 1.65e3·27-s + 5.08e3·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s − 0.223·3-s + 1.09·4-s − 1.07·5-s − 0.323·6-s + 1.11·7-s + 0.137·8-s − 0.949·9-s − 1.55·10-s + 0.301·11-s − 0.244·12-s + 1.01·13-s + 1.61·14-s + 0.239·15-s − 0.896·16-s + 1.54·17-s − 1.37·18-s + 0.232·19-s − 1.17·20-s − 0.250·21-s + 0.436·22-s − 1.78·23-s − 0.0307·24-s + 0.147·25-s + 1.46·26-s + 0.436·27-s + 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.961193867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.961193867\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 8.18T + 32T^{2} \) |
| 3 | \( 1 + 3.48T + 243T^{2} \) |
| 5 | \( 1 + 59.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 145.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 615.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.84e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 366.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.39e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.83e4T + 8.58e9T^{2} \) |
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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
−20.04341950147166817255026287409, −18.18120691900491688818059213142, −16.32063530331324033958351256576, −14.88796235274301749616242188507, −13.97911494013984950872619652115, −12.04382060762040155068358098121, −11.35783285082717437751042033488, −8.097847301882656318990005265289, −5.65524928452027946393136147035, −3.83700042132674970157589138222,
3.83700042132674970157589138222, 5.65524928452027946393136147035, 8.097847301882656318990005265289, 11.35783285082717437751042033488, 12.04382060762040155068358098121, 13.97911494013984950872619652115, 14.88796235274301749616242188507, 16.32063530331324033958351256576, 18.18120691900491688818059213142, 20.04341950147166817255026287409
