L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.900 + 0.433i)13-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 + 0.781i)20-s + (−0.623 + 0.781i)22-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + (−0.623 − 0.781i)29-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.900 + 0.433i)13-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 + 0.781i)20-s + (−0.623 + 0.781i)22-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + (−0.623 − 0.781i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5311338825 + 0.3789968541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5311338825 + 0.3789968541i\) |
\(L(1)\) |
\(\approx\) |
\(0.7124731069 - 0.1218466710i\) |
\(L(1)\) |
\(\approx\) |
\(0.7124731069 - 0.1218466710i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.826 - 0.563i)T \) |
| 5 | \( 1 + (0.955 - 0.294i)T \) |
| 11 | \( 1 + (0.0747 - 0.997i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| 17 | \( 1 + (-0.988 + 0.149i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.623 - 0.781i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.365 + 0.930i)T \) |
| 41 | \( 1 + (0.222 - 0.974i)T \) |
| 43 | \( 1 + (0.222 + 0.974i)T \) |
| 47 | \( 1 + (-0.826 - 0.563i)T \) |
| 53 | \( 1 + (0.365 + 0.930i)T \) |
| 59 | \( 1 + (-0.955 - 0.294i)T \) |
| 61 | \( 1 + (-0.365 + 0.930i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.826 - 0.563i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.0747 + 0.997i)T \) |
| 97 | \( 1 - T \) |
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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
−18.347383579564634383457760406678, −17.90057070682529676895313955020, −17.435479892874376804011871162840, −16.83618440157807656714257202946, −15.97567730098515241012732071257, −15.16065039448997465858654280787, −14.76536101240397109759462572655, −14.0179905142271976441776554756, −13.20238249346209518930352057535, −12.50886774374706329070133372632, −11.380197729563182015484183939191, −10.81670245256034824744504850456, −10.009997350905903290043778179672, −9.43256609909556540696837341559, −9.04607069269824528215015099023, −7.93139915698883418620431926400, −7.084558463231587222046563460889, −6.85453722755850048240962386225, −5.80005885228323598548001357859, −5.17088007992427833954470384673, −4.47265974882924181728991303091, −2.96159808526540546018753033466, −2.18964759767947600235420270935, −1.6208413810550688908320980098, −0.252635368982211754421763519,
1.06466550993486706956215799758, 1.846947931379913216223556063486, 2.553732357179811663540154424, 3.41317454220595841348507613954, 4.34496308650542767148991704417, 5.297428789697639643842023681735, 6.18506256469675659415000115862, 6.84992977059831860887654788406, 7.7737097014749690441912226798, 8.53858873690649047009666149787, 9.19572930672575167023812774077, 9.70811535183353108853818813226, 10.48789644757628390426939228593, 11.09860402571970536424272818473, 11.95021327477632536871680517646, 12.52386776601641903134185786327, 13.41025815475187647256093600299, 13.85421490080131764382867294719, 14.768976043814984441146276737575, 15.79059251418510262294504660702, 16.500342643759607607708340723519, 16.9940607144554583448927962548, 17.5605964310261371904337700939, 18.35517292126324875869981705450, 18.83185018796609629471691960984