L(s) = 1 | + (0.742 − 0.669i)2-s + (−0.536 + 0.843i)3-s + (0.103 − 0.994i)4-s + (0.260 + 0.965i)5-s + (0.166 + 0.986i)6-s + (−0.481 + 0.876i)7-s + (−0.589 − 0.808i)8-s + (−0.424 − 0.905i)9-s + (0.839 + 0.543i)10-s + (0.395 − 0.918i)11-s + (0.783 + 0.620i)12-s + (0.721 + 0.692i)13-s + (0.229 + 0.973i)14-s + (−0.954 − 0.298i)15-s + (−0.978 − 0.205i)16-s + (−0.467 − 0.884i)17-s + ⋯ |
L(s) = 1 | + (0.742 − 0.669i)2-s + (−0.536 + 0.843i)3-s + (0.103 − 0.994i)4-s + (0.260 + 0.965i)5-s + (0.166 + 0.986i)6-s + (−0.481 + 0.876i)7-s + (−0.589 − 0.808i)8-s + (−0.424 − 0.905i)9-s + (0.839 + 0.543i)10-s + (0.395 − 0.918i)11-s + (0.783 + 0.620i)12-s + (0.721 + 0.692i)13-s + (0.229 + 0.973i)14-s + (−0.954 − 0.298i)15-s + (−0.978 − 0.205i)16-s + (−0.467 − 0.884i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9879265880 + 1.155604161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9879265880 + 1.155604161i\) |
\(L(1)\) |
\(\approx\) |
\(1.198016889 + 0.1364746853i\) |
\(L(1)\) |
\(\approx\) |
\(1.198016889 + 0.1364746853i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4729 | \( 1 \) |
good | 2 | \( 1 + (0.742 - 0.669i)T \) |
| 3 | \( 1 + (-0.536 + 0.843i)T \) |
| 5 | \( 1 + (0.260 + 0.965i)T \) |
| 7 | \( 1 + (-0.481 + 0.876i)T \) |
| 11 | \( 1 + (0.395 - 0.918i)T \) |
| 13 | \( 1 + (0.721 + 0.692i)T \) |
| 17 | \( 1 + (-0.467 - 0.884i)T \) |
| 19 | \( 1 + (0.522 + 0.852i)T \) |
| 23 | \( 1 + (-0.675 + 0.737i)T \) |
| 29 | \( 1 + (-0.742 - 0.669i)T \) |
| 31 | \( 1 + (0.639 - 0.768i)T \) |
| 37 | \( 1 + (0.601 + 0.798i)T \) |
| 41 | \( 1 + (-0.651 - 0.758i)T \) |
| 43 | \( 1 + (0.830 + 0.556i)T \) |
| 47 | \( 1 + (-0.915 + 0.402i)T \) |
| 53 | \( 1 + (0.663 - 0.748i)T \) |
| 59 | \( 1 + (0.927 - 0.373i)T \) |
| 61 | \( 1 + (-0.135 + 0.990i)T \) |
| 67 | \( 1 + (0.967 + 0.252i)T \) |
| 71 | \( 1 + (0.939 + 0.343i)T \) |
| 73 | \( 1 + (0.290 + 0.956i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.732 + 0.681i)T \) |
| 89 | \( 1 + (0.0557 + 0.998i)T \) |
| 97 | \( 1 + (-0.481 + 0.876i)T \) |
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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
−17.82415032356947407621585056285, −17.15312402848163901706061102653, −16.73248067404582624036120697989, −16.03942151102122063705719657564, −15.45653147506431848740831651981, −14.43404879134878307854147091516, −13.76526026616939831963793624000, −13.17916498983082380474323540900, −12.73365717315668929357371926351, −12.3298263973465351425934352490, −11.417710636044912427076759245322, −10.69336682320929164932208883421, −9.78458753485485836078305818998, −8.77078097307185108691207656959, −8.22405720077897176240347242160, −7.428237007759190543786709312731, −6.82528008255962094182265361324, −6.20623605537617627944889127146, −5.56171411206705073022598194656, −4.739574409171289199090581502389, −4.20855568821587611051749225023, −3.2969635359351352626544940035, −2.19326887903128040905767295684, −1.38273868943725560337507979591, −0.35971552292813593834021973835,
1.0528418593083362926062018697, 2.22172230445010476122423672336, 2.8703701039683762314805578202, 3.73177314705186214533346040171, 3.96165541996457443664282756676, 5.25557410634408633384857036753, 5.75908196519309636894122287927, 6.29536942047793655657181458851, 6.81665847314304782383243848022, 8.2378781520599507355359979733, 9.28933081954541836550878632651, 9.62567080285650190913395435018, 10.22781345518758804357527223943, 11.22262015569239724478197565730, 11.57051727935628589515506978563, 11.81829032868170038775121284976, 13.0359043202943219666610871842, 13.72694780019028649284286945975, 14.27649324763257493419255265563, 14.932544241261310890416894175972, 15.74535649234120665767549299577, 15.96365381656971800091530513563, 16.86215867850101973915451208122, 17.91928295720859482364320113246, 18.506591416196819483832219677528