L(s) = 1 | + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (−0.900 + 0.433i)5-s + (−0.900 + 0.433i)8-s + (0.0747 + 0.997i)10-s + (0.623 + 0.781i)11-s + (0.365 − 0.930i)13-s + (0.0747 + 0.997i)16-s + (−0.733 + 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.955 + 0.294i)20-s + (0.955 − 0.294i)22-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.733 − 0.680i)26-s + ⋯ |
L(s) = 1 | + (0.365 − 0.930i)2-s + (−0.733 − 0.680i)4-s + (−0.900 + 0.433i)5-s + (−0.900 + 0.433i)8-s + (0.0747 + 0.997i)10-s + (0.623 + 0.781i)11-s + (0.365 − 0.930i)13-s + (0.0747 + 0.997i)16-s + (−0.733 + 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.955 + 0.294i)20-s + (0.955 − 0.294i)22-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.733 − 0.680i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.054650633 + 0.007513239753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054650633 + 0.007513239753i\) |
\(L(1)\) |
\(\approx\) |
\(0.9423004921 - 0.2603904020i\) |
\(L(1)\) |
\(\approx\) |
\(0.9423004921 - 0.2603904020i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.365 - 0.930i)T \) |
| 5 | \( 1 + (-0.900 + 0.433i)T \) |
| 11 | \( 1 + (0.623 + 0.781i)T \) |
| 13 | \( 1 + (0.365 - 0.930i)T \) |
| 17 | \( 1 + (-0.733 + 0.680i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.222 + 0.974i)T \) |
| 29 | \( 1 + (0.955 + 0.294i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.955 + 0.294i)T \) |
| 41 | \( 1 + (0.826 + 0.563i)T \) |
| 43 | \( 1 + (0.826 - 0.563i)T \) |
| 47 | \( 1 + (0.365 - 0.930i)T \) |
| 53 | \( 1 + (0.955 - 0.294i)T \) |
| 59 | \( 1 + (0.826 - 0.563i)T \) |
| 61 | \( 1 + (-0.733 + 0.680i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.988 - 0.149i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.365 + 0.930i)T \) |
| 89 | \( 1 + (0.365 + 0.930i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−24.28246620695590916186817222295, −23.34496788507438627818450147311, −22.53705257277855697421750841795, −21.69986004584339146152155375918, −20.7841432527115339738310971777, −19.65000330306182122953129884803, −18.86424630917614099498204177455, −17.86270692426933700339475649811, −16.76563786953947458287478409988, −16.23354350322214056149408043030, −15.49004374627474937398170223208, −14.49530588337035575919009432388, −13.688193649539976125687967633438, −12.76701651575495496649028489612, −11.78392568740756664560643491590, −11.04863325779220108420476434466, −9.17276925608914843127559191858, −8.80205604673566687333881972406, −7.736254093877165309391745955413, −6.774625324293181529225613008550, −5.95175859086214077458024121438, −4.462445557153488672438855272048, −4.18552488559391476661715072077, −2.75887729144551046593108413642, −0.61147512290570468832534893799,
1.264027655100986308459354253095, 2.56913508659756624488263670740, 3.71798220174239509264929352407, 4.29438404424193059256140538952, 5.62166590837701770390529918299, 6.73619297039492561424310702904, 7.992998882797738014161032473959, 8.909229979895223153514855429463, 10.16307882184793244842551656619, 10.77587016127272212871084427695, 11.77137294362767333774457052202, 12.447745175712483747909014100231, 13.32240969205455452833411086006, 14.56801496392720610271428984646, 15.03751508607473397364409588988, 16.02172688759276438230571290401, 17.52098458662453742165853305897, 18.121557387341076942364669744757, 19.228746168955726763509330424214, 19.81156891771777761185562535083, 20.43329046628195114206443976442, 21.60609973684829152838222942002, 22.317319824169941697303808390436, 23.20531484967475764035617969764, 23.54256121804822566531185926222