L(s) = 1 | + (0.686 − 0.727i)2-s + (−0.0581 − 0.998i)4-s + (0.993 + 0.116i)5-s + (0.893 − 0.448i)7-s + (−0.766 − 0.642i)8-s + (0.766 − 0.642i)10-s + (−0.396 + 0.918i)11-s + (0.973 − 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.993 + 0.116i)16-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.0581 − 0.998i)20-s + (0.396 + 0.918i)22-s + (−0.893 − 0.448i)23-s + ⋯ |
L(s) = 1 | + (0.686 − 0.727i)2-s + (−0.0581 − 0.998i)4-s + (0.993 + 0.116i)5-s + (0.893 − 0.448i)7-s + (−0.766 − 0.642i)8-s + (0.766 − 0.642i)10-s + (−0.396 + 0.918i)11-s + (0.973 − 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.993 + 0.116i)16-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.0581 − 0.998i)20-s + (0.396 + 0.918i)22-s + (−0.893 − 0.448i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.107106244 - 1.949684296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.107106244 - 1.949684296i\) |
\(L(1)\) |
\(\approx\) |
\(1.618427994 - 0.9056868716i\) |
\(L(1)\) |
\(\approx\) |
\(1.618427994 - 0.9056868716i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.686 - 0.727i)T \) |
| 5 | \( 1 + (0.993 + 0.116i)T \) |
| 7 | \( 1 + (0.893 - 0.448i)T \) |
| 11 | \( 1 + (-0.396 + 0.918i)T \) |
| 13 | \( 1 + (0.973 - 0.230i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.893 - 0.448i)T \) |
| 29 | \( 1 + (0.286 + 0.957i)T \) |
| 31 | \( 1 + (-0.835 + 0.549i)T \) |
| 37 | \( 1 + (-0.939 - 0.342i)T \) |
| 41 | \( 1 + (0.686 + 0.727i)T \) |
| 43 | \( 1 + (0.597 + 0.802i)T \) |
| 47 | \( 1 + (0.835 + 0.549i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.396 - 0.918i)T \) |
| 61 | \( 1 + (-0.0581 + 0.998i)T \) |
| 67 | \( 1 + (-0.286 + 0.957i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.686 + 0.727i)T \) |
| 83 | \( 1 + (0.686 - 0.727i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.993 + 0.116i)T \) |
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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
−31.10516220359868132228959230959, −30.16010975498660174662513943913, −29.06305562607839197476821128007, −27.7252178152504595715873396119, −26.36170610024523076994022417828, −25.514448479569282810459371923331, −24.444520485745381868306379402464, −23.77213635188537395279032600420, −22.28562382980713243681153443712, −21.31834705108973595129495204887, −20.79258528578100210335787438524, −18.60436494297736017426025327176, −17.65867984131688942140664279913, −16.60678442544208042213153238469, −15.4207386499413830148193487753, −14.153261517630565279595445187603, −13.48535280733684881744985454359, −12.12821747048661924804010665683, −10.75047072932560272992738109965, −8.88633722057483845053778655391, −7.97154210078463746095180764839, −6.09773321236418388722299949259, −5.51545124185131524715381398673, −3.84046721260376598415495361839, −2.00078796799759549938461208327,
1.323282850427507990554958103325, 2.637960689681680118548723315457, 4.47143921374918441994420529926, 5.5262019200808615016911638096, 7.036338781840297695364750657682, 9.04197289101916887936745111176, 10.3199914924519273068502816932, 11.171591978196345202538129213205, 12.65556419790930653522462179869, 13.728384886403399584268737648928, 14.48477318180073022756282988572, 15.88114582972065334208771313046, 17.79816513339196796030092888669, 18.23374502131793768508457348818, 20.11469131127612700402930019846, 20.72653220080552851140016662160, 21.71762215473169495203786614091, 22.82340360804161002348031078821, 23.85572072699129567125402998304, 24.95458977886989231078564168612, 26.18211315912644746651723984044, 27.68537814799999714097686393658, 28.512342501887479683407637086174, 29.62714388633109677570489635641, 30.44301539641359780216486050780