L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 + 0.642i)5-s + 7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)10-s + (−0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (−0.5 + 0.866i)20-s + (0.173 − 0.984i)22-s + (0.173 + 0.984i)23-s + (0.173 + 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 + 0.642i)5-s + 7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)10-s + (−0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (−0.5 + 0.866i)20-s + (0.173 − 0.984i)22-s + (0.173 + 0.984i)23-s + (0.173 + 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.472200473 + 1.216435073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.472200473 + 1.216435073i\) |
\(L(1)\) |
\(\approx\) |
\(1.500681210 + 0.7922823756i\) |
\(L(1)\) |
\(\approx\) |
\(1.500681210 + 0.7922823756i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 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
−27.79874439033137979567648160509, −26.37684899882191953790535485504, −25.05208295202711793587170514892, −24.26650025693729714381579114913, −23.573525321060139349054725922399, −22.253475008502322703403414471812, −21.425614022267980676973220738, −20.682105537392881935225611710147, −19.94510427853357440652433796832, −18.57508204095415495657627722541, −17.6014232544762497018115381215, −16.53173188493076649487278226851, −14.89435486892500376601372573564, −14.49171213007842294190072866976, −13.08415467655109319685810695852, −12.51717052027169735132551486306, −11.29791445473605933851113038718, −10.19398781328399496334336006289, −9.28433536458382612535871938728, −7.77000633131316166152315280567, −6.19071365525048472641954343942, −5.02061680431562160756247854741, −4.385608332278435621357345655876, −2.42702326291407477151345182173, −1.554927701735068806749286874511,
2.21144956629988592320317634973, 3.3475781889855535376032335453, 5.04953921886665642576142926303, 5.65713293703447488713160801912, 7.09234043984372671332035011689, 7.90735740082866170335182269786, 9.31650602979668311963258297435, 10.81987547412759821768728882816, 11.709901691469754278315281825454, 13.09594979927897328215277718292, 14.01174992859948012014356779908, 14.65339261703294203434325022939, 15.70556489150432348216331878989, 16.97158327104181650666379290538, 17.732127404160690251521426949859, 18.66781440996737261150740263601, 20.35187153023610328237281503849, 21.34788581998815000351172994458, 21.89000220691364266484380380312, 22.93745262428877926058188362774, 24.01474666963255795984805939685, 24.767245676354059429046745897608, 25.61147659921150511080821986849, 26.71956120302593977799755080808, 27.31204833003230620802197920658