| L(s) = 1 | + (0.0680 + 1.41i)2-s + (−1.99 + 0.192i)4-s + (−2.23 − 0.394i)5-s + (−0.842 − 1.00i)7-s + (−0.407 − 2.79i)8-s + (0.405 − 3.18i)10-s + (−0.469 − 2.66i)11-s + (−4.27 + 1.55i)13-s + (1.36 − 1.25i)14-s + (3.92 − 0.765i)16-s + (−4.20 − 2.42i)17-s + (5.44 − 3.14i)19-s + (4.52 + 0.355i)20-s + (3.73 − 0.844i)22-s + (−1.94 − 1.63i)23-s + ⋯ |
| L(s) = 1 | + (0.0481 + 0.998i)2-s + (−0.995 + 0.0961i)4-s + (−1.00 − 0.176i)5-s + (−0.318 − 0.379i)7-s + (−0.143 − 0.989i)8-s + (0.128 − 1.00i)10-s + (−0.141 − 0.803i)11-s + (−1.18 + 0.431i)13-s + (0.363 − 0.336i)14-s + (0.981 − 0.191i)16-s + (−1.02 − 0.589i)17-s + (1.25 − 0.721i)19-s + (1.01 + 0.0794i)20-s + (0.795 − 0.180i)22-s + (−0.405 − 0.340i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00655 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00655 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.202033 - 0.203362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.202033 - 0.203362i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0680 - 1.41i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.23 + 0.394i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.842 + 1.00i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.469 + 2.66i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (4.27 - 1.55i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (4.20 + 2.42i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.44 + 3.14i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.94 + 1.63i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.68 - 4.62i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.87 - 2.23i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.86 - 6.68i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.35 + 3.73i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (8.37 - 1.47i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-9.17 + 7.69i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 3.84iT - 53T^{2} \) |
| 59 | \( 1 + (1.99 - 11.2i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.0301 + 0.0252i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.08 + 11.2i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.22 + 2.12i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.09 - 5.35i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.06 - 5.66i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (8.97 + 3.26i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.99 + 3.45i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.52 + 8.62i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.58087457267894118895120956298, −10.29763373957027703526928019834, −9.217133784534519039938949821483, −8.443253266460654352621112457387, −7.31929589609996164410058618851, −6.87520982591040774104946724005, −5.34670598280060814170169140085, −4.44663130231539304232978982732, −3.26656079138473586696507465402, −0.19557922501564378332673364927,
2.18421942280723539449987561413, 3.49137037275031708755285341512, 4.49882009591324822018459261226, 5.67129340440977214291832223387, 7.35155940887236064908444311880, 8.083902297548685670699975957428, 9.370273358201984757084858758222, 10.00450278364639089545159711637, 11.06528936345329620412447189531, 11.94322450357011451928104380393
