L(s) = 1 | + 2-s + (−0.286 − 0.957i)3-s + 4-s + (−0.893 + 0.448i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s + 8-s + (−0.835 + 0.549i)9-s + (−0.893 + 0.448i)10-s + (−0.893 − 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (0.686 + 0.727i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + 2-s + (−0.286 − 0.957i)3-s + 4-s + (−0.893 + 0.448i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s + 8-s + (−0.835 + 0.549i)9-s + (−0.893 + 0.448i)10-s + (−0.893 − 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (0.686 + 0.727i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2286053108 - 0.8221977912i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2286053108 - 0.8221977912i\) |
\(L(1)\) |
\(\approx\) |
\(1.054341595 - 0.5307012829i\) |
\(L(1)\) |
\(\approx\) |
\(1.054341595 - 0.5307012829i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.286 - 0.957i)T \) |
| 5 | \( 1 + (-0.893 + 0.448i)T \) |
| 7 | \( 1 + (-0.835 - 0.549i)T \) |
| 11 | \( 1 + (-0.893 - 0.448i)T \) |
| 13 | \( 1 + (0.893 - 0.448i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.835 + 0.549i)T \) |
| 31 | \( 1 + (-0.597 + 0.802i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.286 - 0.957i)T \) |
| 53 | \( 1 + (-0.973 - 0.230i)T \) |
| 59 | \( 1 + (-0.686 - 0.727i)T \) |
| 61 | \( 1 + (-0.396 + 0.918i)T \) |
| 67 | \( 1 + (0.286 - 0.957i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.835 + 0.549i)T \) |
| 79 | \( 1 + (-0.686 - 0.727i)T \) |
| 83 | \( 1 + (-0.993 + 0.116i)T \) |
| 89 | \( 1 + (0.0581 - 0.998i)T \) |
| 97 | \( 1 + (-0.597 - 0.802i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.05999190082551447461280604649, −18.227554705387660496623767098965, −17.205890963624631362810472992403, −16.30532669248178189619382137615, −16.02533090467892806880544096607, −15.54996062229783830342405363323, −15.047815526893284539788112654813, −14.16961483065528670882598891463, −13.31287341447227474960705728873, −12.61474962070924266604285928737, −12.10580706799419805986420862410, −11.37633947998500896048740946754, −10.839674259344854013580994304713, −9.9714528540111361100199581476, −9.3403584010841170915483425747, −8.28236174175684331137686266741, −7.792639061321055070966604520878, −6.61869829858684637066980966032, −5.93973776266200388397744165667, −5.4454463964626923592792518155, −4.42032936136930355589191413777, −4.07803495424527423347910582969, −3.31448219533063139115900854551, −2.62297748018390560475257958362, −1.42663992986271528934855288655,
0.18082850360306000997606784182, 1.08976733452429358244689077689, 2.38107713940926746412704528202, 3.08090812270667895525537044876, 3.496488201416678495867368203294, 4.58454823043492869027534007887, 5.37094204946362159217249417658, 6.16299776182644312128096394723, 6.824707868063508313433502829244, 7.30309221027064469077181767255, 7.963467018845257829082576038783, 8.74879021114578981666118176250, 10.18963235670868869839333064454, 10.82981740103867407977922683449, 11.25922086193128540397790562361, 12.082741285843641813426817501789, 12.62283543292235589693782747041, 13.3590077126768705421758670650, 13.79448989178563527731197859271, 14.37088356463355587839472741736, 15.560963984313780044535585811558, 16.00536499852102273871893356966, 16.27040494595466807498970565513, 17.4120168415052320092982866458, 18.36560393370135710093711172911