| L(s) = 1 | + (−7.79 + 1.37i)3-s + (0.638 − 0.535i)5-s + (23.1 − 40.1i)7-s + (−17.1 + 6.25i)9-s + (95.5 + 165. i)11-s + (208. + 36.7i)13-s + (−4.24 + 5.05i)15-s + (504. + 183. i)17-s + (130. − 336. i)19-s + (−125. + 344. i)21-s + (−238. − 200. i)23-s + (−108. + 614. i)25-s + (680. − 393. i)27-s + (67.5 + 185. i)29-s + (−677. − 391. i)31-s + ⋯ |
| L(s) = 1 | + (−0.866 + 0.152i)3-s + (0.0255 − 0.0214i)5-s + (0.472 − 0.818i)7-s + (−0.212 + 0.0771i)9-s + (0.789 + 1.36i)11-s + (1.23 + 0.217i)13-s + (−0.0188 + 0.0224i)15-s + (1.74 + 0.635i)17-s + (0.362 − 0.932i)19-s + (−0.284 + 0.781i)21-s + (−0.450 − 0.378i)23-s + (−0.173 + 0.983i)25-s + (0.934 − 0.539i)27-s + (0.0803 + 0.220i)29-s + (−0.705 − 0.407i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.32749 + 0.256281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32749 + 0.256281i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-130. + 336. i)T \) |
| good | 3 | \( 1 + (7.79 - 1.37i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-0.638 + 0.535i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-23.1 + 40.1i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-95.5 - 165. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-208. - 36.7i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-504. - 183. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (238. + 200. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-67.5 - 185. i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (677. + 391. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.10e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.01e3 + 178. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-867. + 727. i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (1.48e3 - 541. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-623. + 742. i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-1.65e3 + 4.55e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (3.86e3 + 3.24e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (70.2 + 192. i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-3.44e3 - 4.10e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-1.41e3 - 8.04e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-5.64e3 + 995. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (2.38e3 - 4.13e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.10e4 - 1.94e3i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-613. + 1.68e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−13.91627593828646106229576606722, −12.56471888940673237506274329607, −11.51754881607767398050340300969, −10.69179799043532483311705187843, −9.514079133632512755356996955043, −7.916210268474413239999464281291, −6.63022418237021035841818953926, −5.28073933002700703265852554646, −3.92484461329498489741489043521, −1.26031726551423977762649939881,
1.01805101086816540227444021817, 3.43562835380515291680564434257, 5.62095639196438415683661166973, 6.04371930458049287806126178442, 7.995034428941278685004491855207, 9.054474548958259036059572270041, 10.64968024798474271484588313675, 11.68904900056284238618824324441, 12.17038795972245154976691061534, 13.84246762084509965961783450664
