| L(s) = 1 | + 3.98·2-s − 20.3·3-s − 16.1·4-s − 25·5-s − 80.9·6-s + 158.·7-s − 191.·8-s + 170.·9-s − 99.5·10-s − 121·11-s + 328.·12-s − 696.·13-s + 630.·14-s + 508.·15-s − 246.·16-s − 2.16e3·17-s + 677.·18-s + 361·19-s + 403.·20-s − 3.21e3·21-s − 481.·22-s + 2.91e3·23-s + 3.89e3·24-s + 625·25-s − 2.77e3·26-s + 1.48e3·27-s − 2.55e3·28-s + ⋯ |
| L(s) = 1 | + 0.703·2-s − 1.30·3-s − 0.504·4-s − 0.447·5-s − 0.917·6-s + 1.22·7-s − 1.05·8-s + 0.700·9-s − 0.314·10-s − 0.301·11-s + 0.657·12-s − 1.14·13-s + 0.859·14-s + 0.583·15-s − 0.240·16-s − 1.81·17-s + 0.492·18-s + 0.229·19-s + 0.225·20-s − 1.59·21-s − 0.212·22-s + 1.14·23-s + 1.38·24-s + 0.200·25-s − 0.804·26-s + 0.390·27-s − 0.616·28-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
| \(L(3)\) |
\(\approx\) |
\(0.2214640134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2214640134\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 3.98T + 32T^{2} \) |
| 3 | \( 1 + 20.3T + 243T^{2} \) |
| 7 | \( 1 - 158.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 696.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.16e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.91e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.56e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.59e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 389.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.39e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 911.T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.62e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.20e5T + 8.58e9T^{2} \) |
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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
−9.063907155921012170819942354624, −8.431210009589293438663229186651, −7.26354267646581501467513694564, −6.53874368736489825004943610930, −5.30685749185993426908374612699, −4.94939504569454534025019797082, −4.44265477621157942018452483314, −3.10227910717787618664515074142, −1.69412719028216158630036977339, −0.19342058553487729534264911316,
0.19342058553487729534264911316, 1.69412719028216158630036977339, 3.10227910717787618664515074142, 4.44265477621157942018452483314, 4.94939504569454534025019797082, 5.30685749185993426908374612699, 6.53874368736489825004943610930, 7.26354267646581501467513694564, 8.431210009589293438663229186651, 9.063907155921012170819942354624
