| L(s) = 1 | + 14.6·2-s + (24.9 + 43.2i)3-s + 86.9·4-s + (29.8 + 51.6i)5-s + (365. + 633. i)6-s + (−872. + 1.51e3i)7-s − 602.·8-s + (−152. + 264. i)9-s + (437. + 757. i)10-s + 5.70e3·11-s + (2.17e3 + 3.75e3i)12-s + (−549. + 951. i)13-s + (−1.27e4 + 2.21e4i)14-s + (−1.48e3 + 2.57e3i)15-s − 1.99e4·16-s + (9.55e3 − 1.65e4i)17-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + (0.533 + 0.924i)3-s + 0.679·4-s + (0.106 + 0.184i)5-s + (0.691 + 1.19i)6-s + (−0.961 + 1.66i)7-s − 0.415·8-s + (−0.0698 + 0.121i)9-s + (0.138 + 0.239i)10-s + 1.29·11-s + (0.362 + 0.627i)12-s + (−0.0693 + 0.120i)13-s + (−1.24 + 2.15i)14-s + (−0.113 + 0.197i)15-s − 1.21·16-s + (0.471 − 0.817i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.41471 + 2.75925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.41471 + 2.75925i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.98e5 + 4.27e5i)T \) |
| good | 2 | \( 1 - 14.6T + 128T^{2} \) |
| 3 | \( 1 + (-24.9 - 43.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-29.8 - 51.6i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (872. - 1.51e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 5.70e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (549. - 951. i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-9.55e3 + 1.65e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-5.16e3 - 8.94e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-5.46e4 - 9.47e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.67e3 - 2.90e3i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (9.55e4 + 1.65e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.28e5 - 2.23e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 881.T + 1.94e11T^{2} \) |
| 47 | \( 1 - 5.59e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (6.06e5 + 1.05e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 4.21e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + (9.76e5 - 1.69e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.22e6 + 2.11e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (4.63e5 - 8.02e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.81e6 + 4.87e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.01e6 - 3.48e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.16e6 - 7.21e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-3.27e6 - 5.67e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.05e7T + 8.07e13T^{2} \) |
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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
−14.88002606747384056113871132500, −13.80059202169148390807844002862, −12.44621140765586130572357734897, −11.68694710742402685074207413745, −9.482868201144144814912070868856, −9.131021543949694952893356849394, −6.49028753245017684925809886947, −5.32520148744247222883479468939, −3.75702502133534904882719330932, −2.81009732478636348596847766663,
1.05212298648497453463964956551, 3.18345379851580519665533409893, 4.37412529906936773001867941074, 6.40931277425921863271342201331, 7.23612594294344946926946752562, 9.074024029054718870567150606661, 10.74400450121496450839931862630, 12.57685950891824709593457738903, 12.96602065711819558531240112651, 14.00935335478405332759224772052
