| L(s) = 1 | + 10.1·2-s + 30.5·3-s + 71.8·4-s + 25·5-s + 310.·6-s + 96.2·7-s + 406.·8-s + 687.·9-s + 254.·10-s + 121·11-s + 2.19e3·12-s − 1.15e3·13-s + 980.·14-s + 762.·15-s + 1.84e3·16-s + 322.·17-s + 7.01e3·18-s + 361·19-s + 1.79e3·20-s + 2.93e3·21-s + 1.23e3·22-s + 1.02e3·23-s + 1.24e4·24-s + 625·25-s − 1.17e4·26-s + 1.35e4·27-s + 6.91e3·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 1.95·3-s + 2.24·4-s + 0.447·5-s + 3.52·6-s + 0.742·7-s + 2.24·8-s + 2.83·9-s + 0.805·10-s + 0.301·11-s + 4.39·12-s − 1.89·13-s + 1.33·14-s + 0.875·15-s + 1.80·16-s + 0.270·17-s + 5.10·18-s + 0.229·19-s + 1.00·20-s + 1.45·21-s + 0.543·22-s + 0.402·23-s + 4.39·24-s + 0.200·25-s − 3.40·26-s + 3.58·27-s + 1.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(18.86453916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.86453916\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 10.1T + 32T^{2} \) |
| 3 | \( 1 - 30.5T + 243T^{2} \) |
| 7 | \( 1 - 96.2T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 322.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 981.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.42e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.83e4T + 8.58e9T^{2} \) |
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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
−9.228768470058928528622897568236, −8.133996016742353962814829528773, −7.29984922982918480850373218449, −6.87101767543977921280736542944, −5.30194366790276337715335866894, −4.75200848353484938891158044498, −3.78726049447521802975763928689, −3.04835848879158217622665488729, −2.17958890106075717700026901303, −1.67088391911289417967939793856,
1.67088391911289417967939793856, 2.17958890106075717700026901303, 3.04835848879158217622665488729, 3.78726049447521802975763928689, 4.75200848353484938891158044498, 5.30194366790276337715335866894, 6.87101767543977921280736542944, 7.29984922982918480850373218449, 8.133996016742353962814829528773, 9.228768470058928528622897568236
