L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.939 + 0.342i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 + 0.342i)13-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 18-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.939 + 0.342i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 + 0.342i)13-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.052529773 + 2.813019455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.052529773 + 2.813019455i\) |
\(L(1)\) |
\(\approx\) |
\(1.754307678 + 1.134240393i\) |
\(L(1)\) |
\(\approx\) |
\(1.754307678 + 1.134240393i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.939 + 0.342i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (0.766 - 0.642i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
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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
−28.18549611985563238165776552810, −27.27936913985296330303165988817, −26.08023416596582459469203218427, −24.74946576196209656365149847689, −24.19875007347812737869472156241, −23.05352815990100000310272586045, −21.65661652168235781142390603408, −21.02670147651540026110752821489, −20.17856141420813101992822568578, −19.42673268025644707232276819032, −18.30530768121975799621298997136, −16.341024027070239039304876169, −15.634183329540732950922636720097, −14.51251421056240918227964880583, −13.27749513702194469287124092380, −12.9326177461546948048312001685, −11.32363381385143927261952187298, −10.28597934326759304591297592733, −8.97207077818609683938324681363, −8.152157646619107974138329172177, −6.1468152304444000768658121077, −4.78117121426434769324681329709, −3.832608706468378034540666889614, −2.57604114162626871844547717988, −1.02239918999271247624188261185,
2.24637700256513339442010762903, 3.302930732394587956841442665633, 4.48381941620394831551183602805, 6.31137136018499227974859891899, 7.19083385580164522691068778186, 8.074594257025538994873586605059, 9.43441961320949033599330984001, 11.06438980979639245117395843292, 12.3383960963824873476760600507, 13.54680009866454106597968017418, 14.128867027522916045814372475, 15.357115070420711804335591639076, 15.736624797000087124965734525913, 17.659171267363978559806699978430, 18.383413443849533609196320728570, 19.693048436747817765379163767573, 20.79128952038048489923807020094, 21.671454736141589160194233279310, 22.98170923702373068463826873096, 23.54348080128764547089382439286, 24.8385456424169106731722975751, 25.65027677627902574179221665633, 26.28655012139206024811256619921, 27.18410537505975913716297121090, 29.02550908747029692044349577005