| L(s) = 1 | + (−16 − 27.7i)2-s + (94.7 − 164. i)3-s + (−511. + 886. i)4-s + (3.21e3 + 5.56e3i)5-s − 6.06e3·6-s + 3.27e4·8-s + (7.06e4 + 1.22e5i)9-s + (1.02e5 − 1.77e5i)10-s + (4.59e5 − 7.96e5i)11-s + (9.70e4 + 1.68e5i)12-s − 1.22e6·13-s + 1.21e6·15-s + (−5.24e5 − 9.08e5i)16-s + (−3.31e6 + 5.73e6i)17-s + (2.26e6 − 3.91e6i)18-s + (−9.33e6 − 1.61e7i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.225 − 0.389i)3-s + (−0.249 + 0.433i)4-s + (0.459 + 0.795i)5-s − 0.318·6-s + 0.353·8-s + (0.398 + 0.690i)9-s + (0.324 − 0.562i)10-s + (0.861 − 1.49i)11-s + (0.112 + 0.194i)12-s − 0.917·13-s + 0.413·15-s + (−0.125 − 0.216i)16-s + (−0.565 + 0.979i)17-s + (0.281 − 0.488i)18-s + (−0.864 − 1.49i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.86812 - 0.926079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86812 - 0.926079i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 + 27.7i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-94.7 + 164. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-3.21e3 - 5.56e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-4.59e5 + 7.96e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.22e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (3.31e6 - 5.73e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (9.33e6 + 1.61e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.43e7 - 2.48e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 7.73e6T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-5.68e7 + 9.85e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-9.50e7 - 1.64e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.74e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.83e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.75e8 - 9.97e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (3.18e8 - 5.51e8i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (1.39e9 - 2.41e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.60e9 + 4.50e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-3.19e9 + 5.52e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.41e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (8.18e9 - 1.41e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.91e9 - 3.31e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 2.84e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.35e10 + 2.34e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.49e11T + 7.15e21T^{2} \) |
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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
−11.20593466987414413135111705358, −10.74002420229412637127470521584, −9.438871644711413092000346793537, −8.434280542341209984947388135192, −7.19399231865169519960093443690, −6.14589607890326984310752866951, −4.40078084122860573446840334207, −2.92172607628418580498473542499, −2.07270325251161588939534075741, −0.72240207215710175743419931728,
0.858014306242569436966312888429, 2.09771743930133555947293517077, 4.15754515715625086909640194088, 4.94011531809592471017167007669, 6.43997051770267788064546180106, 7.40121712192797425900033621810, 8.928616604941631402832047234593, 9.459275559000562399522057230018, 10.32160767873261158259899352205, 12.17604666064679204943729832137
