L(s) = 1 | + 3.65i·2-s + (15.5 + 0.882i)3-s + 18.6·4-s − 74.3·5-s + (−3.22 + 56.8i)6-s + 185. i·8-s + (241. + 27.4i)9-s − 271. i·10-s − 39.3i·11-s + (290. + 16.4i)12-s + 589. i·13-s + (−1.15e3 − 65.5i)15-s − 80.0·16-s − 1.23e3·17-s + (−100. + 882. i)18-s + 2.76e3i·19-s + ⋯ |
L(s) = 1 | + 0.646i·2-s + (0.998 + 0.0566i)3-s + 0.582·4-s − 1.32·5-s + (−0.0365 + 0.645i)6-s + 1.02i·8-s + (0.993 + 0.113i)9-s − 0.859i·10-s − 0.0981i·11-s + (0.581 + 0.0329i)12-s + 0.967i·13-s + (−1.32 − 0.0752i)15-s − 0.0781·16-s − 1.03·17-s + (−0.0730 + 0.641i)18-s + 1.75i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.165137077\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.165137077\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-15.5 - 0.882i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 3.65iT - 32T^{2} \) |
| 5 | \( 1 + 74.3T + 3.12e3T^{2} \) |
| 11 | \( 1 + 39.3iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 589. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.76e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 463. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.29e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.84e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.84e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.75e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.60e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.74e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.61e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.06e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 5.16e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.71e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.86e4iT - 8.58e9T^{2} \) |
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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
−12.44541654787599634073716003727, −11.58812265645888371164716559183, −10.55431316947307413081129271516, −9.030429152750099148865490259581, −8.101922845119879336636159724674, −7.43409666423081265438318589948, −6.40215689464651764844780710137, −4.51792134069441982557268963226, −3.40864166786711420539604501591, −1.87429403745950027291544769882,
0.62115573056224084981996490456, 2.39687537824437042594724644325, 3.37644800200622311687902675579, 4.48251686094907719670026878219, 6.75003857059878105144238100250, 7.59644433996418790293340182757, 8.553175358039279636346819392560, 9.780130820372798947140485898938, 10.94611559376433363191974979221, 11.65782589516049674974261460743