| L(s) = 1 | − 9.01·2-s + 28.8·3-s + 49.2·4-s − 22.2·5-s − 260.·6-s − 142.·7-s − 155.·8-s + 589.·9-s + 200.·10-s − 143.·11-s + 1.42e3·12-s + 224.·13-s + 1.28e3·14-s − 642.·15-s − 173.·16-s − 592.·17-s − 5.31e3·18-s − 32.3·19-s − 1.09e3·20-s − 4.10e3·21-s + 1.29e3·22-s − 3.08e3·23-s − 4.49e3·24-s − 2.62e3·25-s − 2.02e3·26-s + 9.99e3·27-s − 7.00e3·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.85·3-s + 1.53·4-s − 0.398·5-s − 2.94·6-s − 1.09·7-s − 0.859·8-s + 2.42·9-s + 0.635·10-s − 0.357·11-s + 2.84·12-s + 0.367·13-s + 1.74·14-s − 0.737·15-s − 0.169·16-s − 0.497·17-s − 3.86·18-s − 0.0205·19-s − 0.613·20-s − 2.02·21-s + 0.569·22-s − 1.21·23-s − 1.59·24-s − 0.841·25-s − 0.586·26-s + 2.63·27-s − 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + 3.88e4T \) |
| good | 2 | \( 1 + 9.01T + 32T^{2} \) |
| 3 | \( 1 - 28.8T + 243T^{2} \) |
| 5 | \( 1 + 22.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 142.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 224.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 592.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 32.3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.08e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.34e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 593.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.51e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.52e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.10e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.31e5T + 8.58e9T^{2} \) |
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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
−10.41256204946801471476052648067, −9.855245839517909345411101146779, −8.980773012500813581749188087944, −8.319622810965583184928877400141, −7.54417942628476752259083871690, −6.59056457868262419013004942904, −3.99862284901154521402241232678, −2.86544574427182806128562929993, −1.73652246262565919358673756589, 0,
1.73652246262565919358673756589, 2.86544574427182806128562929993, 3.99862284901154521402241232678, 6.59056457868262419013004942904, 7.54417942628476752259083871690, 8.319622810965583184928877400141, 8.980773012500813581749188087944, 9.855245839517909345411101146779, 10.41256204946801471476052648067
