L(s) = 1 | + (−0.125 + 0.992i)2-s + (−0.248 + 0.968i)3-s + (−0.968 − 0.248i)4-s + (−0.929 − 0.368i)6-s − i·7-s + (0.368 − 0.929i)8-s + (−0.876 − 0.481i)9-s + (0.481 − 0.876i)12-s + (−0.481 + 0.876i)13-s + (0.992 + 0.125i)14-s + (0.876 + 0.481i)16-s + (0.770 + 0.637i)17-s + (0.587 − 0.809i)18-s + (0.637 − 0.770i)19-s + (0.968 + 0.248i)21-s + ⋯ |
L(s) = 1 | + (−0.125 + 0.992i)2-s + (−0.248 + 0.968i)3-s + (−0.968 − 0.248i)4-s + (−0.929 − 0.368i)6-s − i·7-s + (0.368 − 0.929i)8-s + (−0.876 − 0.481i)9-s + (0.481 − 0.876i)12-s + (−0.481 + 0.876i)13-s + (0.992 + 0.125i)14-s + (0.876 + 0.481i)16-s + (0.770 + 0.637i)17-s + (0.587 − 0.809i)18-s + (0.637 − 0.770i)19-s + (0.968 + 0.248i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6286994058 - 0.1599136498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6286994058 - 0.1599136498i\) |
\(L(1)\) |
\(\approx\) |
\(0.6362172813 + 0.4198618588i\) |
\(L(1)\) |
\(\approx\) |
\(0.6362172813 + 0.4198618588i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.125 + 0.992i)T \) |
| 3 | \( 1 + (-0.248 + 0.968i)T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-0.481 + 0.876i)T \) |
| 17 | \( 1 + (0.770 + 0.637i)T \) |
| 19 | \( 1 + (0.637 - 0.770i)T \) |
| 23 | \( 1 + (0.982 - 0.187i)T \) |
| 29 | \( 1 + (-0.0627 - 0.998i)T \) |
| 31 | \( 1 + (0.535 + 0.844i)T \) |
| 37 | \( 1 + (0.982 + 0.187i)T \) |
| 41 | \( 1 + (-0.992 + 0.125i)T \) |
| 43 | \( 1 + (-0.587 - 0.809i)T \) |
| 47 | \( 1 + (-0.844 - 0.535i)T \) |
| 53 | \( 1 + (-0.368 - 0.929i)T \) |
| 59 | \( 1 + (-0.728 + 0.684i)T \) |
| 61 | \( 1 + (-0.425 + 0.904i)T \) |
| 67 | \( 1 + (-0.368 + 0.929i)T \) |
| 71 | \( 1 + (0.0627 + 0.998i)T \) |
| 73 | \( 1 + (0.982 - 0.187i)T \) |
| 79 | \( 1 + (-0.0627 - 0.998i)T \) |
| 83 | \( 1 + (0.998 + 0.0627i)T \) |
| 89 | \( 1 + (-0.876 + 0.481i)T \) |
| 97 | \( 1 + (-0.844 - 0.535i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.56884485807696109610410360553, −19.894429278113959109308852516286, −19.11084782163544380603983709857, −18.39297005769195315692859765398, −18.168695778248444500010640445313, −17.152224696174381211246481017, −16.49486257847288173666547296906, −15.17216152551042084495984959698, −14.39587663903375900209986948195, −13.59128523709666164548739009686, −12.68250325917077505218671615459, −12.342789179037056066402783903908, −11.58158160089293038719583667828, −10.90732204564728686090970741529, −9.79616891131973715560025646458, −9.173310416317205402377398515738, −8.03450875265948145057413142106, −7.73142806669830650801690979028, −6.37408717273344313034773800437, −5.37865531196823900901223676119, −4.95619402029891100741143534403, −3.211801092549144988304499591347, −2.84346488668469370555759815880, −1.74796302829672919948381633579, −0.932184391448842860889378763355,
0.173715223808058401352509367450, 1.227064320491973873351913195691, 3.076780260170316375896732872389, 3.98645862459419636946210153528, 4.6882549157238041001505617123, 5.33747861433096899127683509479, 6.44887393560853587942998658606, 7.05260413473016633227736448311, 8.04213191099047715081441391630, 8.88092901686406883811331492768, 9.765480271940998514375753567800, 10.18053629854103702154945425547, 11.15952658788173203133135752092, 12.059427744023943140540200505257, 13.2885534301628968565075957647, 13.90929510919180420189287812134, 14.73740655089570659809256307467, 15.23203304384037547969284373504, 16.23888164501571915594939994230, 16.741823699494916180244638038427, 17.21581115912176148231776155393, 17.9909698341274568783908711149, 19.1186793472926279611847032592, 19.72655270282909480563955877502, 20.737176054935411330721604805036