| L(s) = 1 | + 2.73·2-s − 29.5·3-s − 24.5·4-s − 25·5-s − 80.9·6-s + 175.·7-s − 154.·8-s + 632.·9-s − 68.3·10-s + 121·11-s + 725.·12-s − 441.·13-s + 481.·14-s + 739.·15-s + 361.·16-s + 340.·17-s + 1.72e3·18-s − 960.·19-s + 612.·20-s − 5.20e3·21-s + 330.·22-s + 1.30e3·23-s + 4.57e3·24-s + 625·25-s − 1.20e3·26-s − 1.15e4·27-s − 4.31e3·28-s + ⋯ |
| L(s) = 1 | + 0.483·2-s − 1.89·3-s − 0.766·4-s − 0.447·5-s − 0.917·6-s + 1.35·7-s − 0.854·8-s + 2.60·9-s − 0.216·10-s + 0.301·11-s + 1.45·12-s − 0.724·13-s + 0.655·14-s + 0.848·15-s + 0.353·16-s + 0.285·17-s + 1.25·18-s − 0.610·19-s + 0.342·20-s − 2.57·21-s + 0.145·22-s + 0.514·23-s + 1.62·24-s + 0.200·25-s − 0.350·26-s − 3.03·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8881749871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8881749871\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 2.73T + 32T^{2} \) |
| 3 | \( 1 + 29.5T + 243T^{2} \) |
| 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 441.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 340.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 960.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.00e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.07e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.88e4T + 8.58e9T^{2} \) |
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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
−14.34036499246723941747289978850, −12.81534700303184943638105894047, −11.95671137919171856141557479934, −11.21224488234256247487713498141, −9.939898502645879939190774206484, −8.065135576433019066855067755115, −6.42800852916647757865537717115, −4.99966082622999785851961049667, −4.48859712253087428608424215645, −0.828451854330644962842995740742,
0.828451854330644962842995740742, 4.48859712253087428608424215645, 4.99966082622999785851961049667, 6.42800852916647757865537717115, 8.065135576433019066855067755115, 9.939898502645879939190774206484, 11.21224488234256247487713498141, 11.95671137919171856141557479934, 12.81534700303184943638105894047, 14.34036499246723941747289978850
