| L(s) = 1 | + 16.3·2-s + (−17.0 + 29.4i)3-s + 140.·4-s + (160. − 277. i)5-s + (−278. + 482. i)6-s + (736. + 1.27e3i)7-s + 210.·8-s + (515. + 892. i)9-s + (2.62e3 − 4.54e3i)10-s + 6.33e3·11-s + (−2.39e3 + 4.14e3i)12-s + (−4.36e3 − 7.56e3i)13-s + (1.20e4 + 2.09e4i)14-s + (5.44e3 + 9.43e3i)15-s − 1.45e4·16-s + (1.69e4 + 2.94e4i)17-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + (−0.363 + 0.629i)3-s + 1.10·4-s + (0.572 − 0.992i)5-s + (−0.526 + 0.912i)6-s + (0.811 + 1.40i)7-s + 0.145·8-s + (0.235 + 0.408i)9-s + (0.830 − 1.43i)10-s + 1.43·11-s + (−0.400 + 0.693i)12-s + (−0.551 − 0.954i)13-s + (1.17 + 2.03i)14-s + (0.416 + 0.721i)15-s − 0.889·16-s + (0.838 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.83613 + 1.33923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.83613 + 1.33923i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-8.13e4 + 5.14e5i)T \) |
| good | 2 | \( 1 - 16.3T + 128T^{2} \) |
| 3 | \( 1 + (17.0 - 29.4i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-160. + 277. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-736. - 1.27e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 6.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (4.36e3 + 7.56e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.69e4 - 2.94e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.93e3 - 3.35e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-7.30e3 + 1.26e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.02e5 + 1.77e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-9.81e4 + 1.70e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.50e5 + 2.61e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 8.13e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 8.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.23e5 + 2.14e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 - 1.91e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.11e6 - 1.93e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (3.64e5 - 6.30e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (4.19e5 + 7.27e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-7.46e5 - 1.29e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.26e5 + 7.38e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.15e6 + 2.00e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (4.37e6 - 7.58e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 3.27e5T + 8.07e13T^{2} \) |
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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
−14.79940790155879878881348896818, −13.27890491755201130175317984560, −12.34563177281438030746241477263, −11.52660872237535255704226634540, −9.750126815513814057908158369462, −8.424081053109623282653770810003, −5.83931545889696770537998744084, −5.28787317850149207515633771180, −4.09081964448607944789538692681, −1.93783579111175780097830528758,
1.41597126906881932907789988388, 3.45694837342230685702861172600, 4.84045394682732888732844789493, 6.71414154608815713555310057659, 6.96543480724185688816584455255, 9.688692990020518325109831999496, 11.32303451927140909832528928467, 11.98347186468056594237390646677, 13.46793863506607213708820407712, 14.32884525932781262372949474274
