L(s) = 1 | + (0.893 + 0.448i)5-s + (−0.0581 + 0.998i)11-s + (−0.597 + 0.802i)13-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.286 + 0.957i)23-s + (0.597 + 0.802i)25-s + (−0.396 + 0.918i)29-s + (−0.686 − 0.727i)31-s + (0.173 + 0.984i)37-s + (−0.993 − 0.116i)41-s + (0.835 + 0.549i)43-s + (0.686 − 0.727i)47-s + (0.5 + 0.866i)53-s + (−0.5 + 0.866i)55-s + ⋯ |
L(s) = 1 | + (0.893 + 0.448i)5-s + (−0.0581 + 0.998i)11-s + (−0.597 + 0.802i)13-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.286 + 0.957i)23-s + (0.597 + 0.802i)25-s + (−0.396 + 0.918i)29-s + (−0.686 − 0.727i)31-s + (0.173 + 0.984i)37-s + (−0.993 − 0.116i)41-s + (0.835 + 0.549i)43-s + (0.686 − 0.727i)47-s + (0.5 + 0.866i)53-s + (−0.5 + 0.866i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4343727437 + 1.915336977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4343727437 + 1.915336977i\) |
\(L(1)\) |
\(\approx\) |
\(1.122191594 + 0.4045075732i\) |
\(L(1)\) |
\(\approx\) |
\(1.122191594 + 0.4045075732i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.893 + 0.448i)T \) |
| 11 | \( 1 + (-0.0581 + 0.998i)T \) |
| 13 | \( 1 + (-0.597 + 0.802i)T \) |
| 17 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.286 + 0.957i)T \) |
| 29 | \( 1 + (-0.396 + 0.918i)T \) |
| 31 | \( 1 + (-0.686 - 0.727i)T \) |
| 37 | \( 1 + (0.173 + 0.984i)T \) |
| 41 | \( 1 + (-0.993 - 0.116i)T \) |
| 43 | \( 1 + (0.835 + 0.549i)T \) |
| 47 | \( 1 + (0.686 - 0.727i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.0581 + 0.998i)T \) |
| 61 | \( 1 + (-0.973 - 0.230i)T \) |
| 67 | \( 1 + (-0.396 - 0.918i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.993 - 0.116i)T \) |
| 83 | \( 1 + (0.993 - 0.116i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.893 + 0.448i)T \) |
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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
−19.199724659999427003347455524426, −18.380079412958556117996322810807, −17.70606645322774055748035445715, −17.02787180754481147560283466987, −16.42395084121870190066875978624, −15.69261278687003612393048090267, −14.699009358986209212095638567673, −14.0973927364872476330447737775, −13.38488403741775391613154705349, −12.6789846198071624986489042578, −12.0685647706533550342106568596, −10.976713051284728417438924389405, −10.35668963233138785446713897378, −9.62480926894186975586834688803, −8.85487146925950594228581188914, −8.14449672491856995835873551904, −7.309565876766924985145371808102, −6.244713293462861083429651999481, −5.5976914856102223986249094410, −5.06333376303681843773428687612, −3.91794500541956524299866986863, −2.97164894688317903927282381380, −2.19642921766331077143891943168, −1.081529959875597016684521304403, −0.32848447868989439263385657526,
1.30491534770324119951865043817, 1.95279167067648582726807628418, 2.861843098722196214811200442254, 3.75335662748240672115787999464, 4.867540951299649476971688849476, 5.47945441013515230867437022228, 6.32867174259131936769110321783, 7.33916086333296385763923844833, 7.54314104576574110573878157491, 9.02760465154753288906073673729, 9.61062638730698415023596240937, 10.05640818956710082320031083865, 10.96803725933493430186175750497, 11.94445338990141227942642603781, 12.37246702693138880447547757335, 13.54674731569572004471830598604, 13.90327246417196753322805954549, 14.785297815337792897643397762331, 15.24060774490380310928169445208, 16.500280274339264041507596098285, 16.81401133361856686211625398018, 17.779019015730862393297293358, 18.324170144722344211453177106382, 18.87795817098360479516945765683, 19.90723336606321209837258282696