| L(s) = 1 | + (−1.07 − 0.916i)2-s + (0.459 + 1.66i)3-s + (0.318 + 1.97i)4-s + (−0.221 + 1.25i)5-s + (1.03 − 2.21i)6-s + (−0.0241 − 0.0287i)7-s + (1.46 − 2.41i)8-s + (−2.57 + 1.53i)9-s + (1.39 − 1.15i)10-s + (−3.96 + 0.699i)11-s + (−3.15 + 1.43i)12-s + (1.52 + 4.18i)13-s + (−0.000387 + 0.0531i)14-s + (−2.20 + 0.208i)15-s + (−3.79 + 1.25i)16-s + (−0.960 − 0.554i)17-s + ⋯ |
| L(s) = 1 | + (−0.761 − 0.648i)2-s + (0.265 + 0.964i)3-s + (0.159 + 0.987i)4-s + (−0.0992 + 0.562i)5-s + (0.422 − 0.906i)6-s + (−0.00912 − 0.0108i)7-s + (0.518 − 0.854i)8-s + (−0.859 + 0.511i)9-s + (0.440 − 0.364i)10-s + (−1.19 + 0.210i)11-s + (−0.909 + 0.415i)12-s + (0.422 + 1.16i)13-s + (−0.000103 + 0.0142i)14-s + (−0.568 + 0.0537i)15-s + (−0.949 + 0.314i)16-s + (−0.232 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0303 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0303 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.544243 + 0.527980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.544243 + 0.527980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 + 0.916i)T \) |
| 3 | \( 1 + (-0.459 - 1.66i)T \) |
| good | 5 | \( 1 + (0.221 - 1.25i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.0241 + 0.0287i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (3.96 - 0.699i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.52 - 4.18i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.960 + 0.554i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 2.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.78 + 1.49i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.19 - 2.98i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.991 + 1.18i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.0149 + 0.00865i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.470 + 1.29i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0865 + 0.490i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.19 - 6.03i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + (-1.54 - 0.271i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (9.01 + 10.7i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-7.92 + 2.88i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.72 + 8.17i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.64 - 11.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.05 + 8.39i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.13 - 11.3i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.90 + 3.98i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0961 + 0.545i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.22893985543574444287209350371, −11.23592265759756916567588621902, −10.52434248482786280300792029406, −9.823432112567945925307268059795, −8.781190945560016522447020589966, −7.928037593679651415795894098759, −6.65451996380925953431374988892, −4.84490818584252670584648688998, −3.58346675987543047747961656570, −2.42281951888259962727640758678,
0.801817883204383452621776080209, 2.68727373923703040205122408325, 5.08620014819092196409342606996, 6.06231403620914632599425444075, 7.23973578937972677591906597477, 8.207079357607071201248541964179, 8.620601158275826761715780847407, 10.00615622932403498028462919538, 10.97234906073843769757003286853, 12.16789403319687522010296126584
