L(s) = 1 | + (0.868 − 0.495i)2-s + (−0.590 − 0.807i)3-s + (0.509 − 0.860i)4-s + (0.898 + 0.438i)5-s + (−0.912 − 0.408i)6-s + (0.948 − 0.318i)7-s + (0.0161 − 0.999i)8-s + (−0.302 + 0.953i)9-s + (0.997 − 0.0647i)10-s + (0.393 − 0.919i)11-s + (−0.995 + 0.0970i)12-s + (−0.641 + 0.767i)13-s + (0.665 − 0.746i)14-s + (−0.177 − 0.984i)15-s + (−0.481 − 0.876i)16-s + (0.997 − 0.0647i)17-s + ⋯ |
L(s) = 1 | + (0.868 − 0.495i)2-s + (−0.590 − 0.807i)3-s + (0.509 − 0.860i)4-s + (0.898 + 0.438i)5-s + (−0.912 − 0.408i)6-s + (0.948 − 0.318i)7-s + (0.0161 − 0.999i)8-s + (−0.302 + 0.953i)9-s + (0.997 − 0.0647i)10-s + (0.393 − 0.919i)11-s + (−0.995 + 0.0970i)12-s + (−0.641 + 0.767i)13-s + (0.665 − 0.746i)14-s + (−0.177 − 0.984i)15-s + (−0.481 − 0.876i)16-s + (0.997 − 0.0647i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0928 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0928 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.534092431 - 1.683794966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.534092431 - 1.683794966i\) |
\(L(1)\) |
\(\approx\) |
\(1.487063977 - 0.9424310964i\) |
\(L(1)\) |
\(\approx\) |
\(1.487063977 - 0.9424310964i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 389 | \( 1 \) |
good | 2 | \( 1 + (0.868 - 0.495i)T \) |
| 3 | \( 1 + (-0.590 - 0.807i)T \) |
| 5 | \( 1 + (0.898 + 0.438i)T \) |
| 7 | \( 1 + (0.948 - 0.318i)T \) |
| 11 | \( 1 + (0.393 - 0.919i)T \) |
| 13 | \( 1 + (-0.641 + 0.767i)T \) |
| 17 | \( 1 + (0.997 - 0.0647i)T \) |
| 19 | \( 1 + (-0.777 + 0.628i)T \) |
| 23 | \( 1 + (0.898 - 0.438i)T \) |
| 29 | \( 1 + (0.712 + 0.701i)T \) |
| 31 | \( 1 + (-0.999 - 0.0323i)T \) |
| 37 | \( 1 + (-0.816 + 0.577i)T \) |
| 41 | \( 1 + (-0.777 - 0.628i)T \) |
| 43 | \( 1 + (-0.735 - 0.677i)T \) |
| 47 | \( 1 + (-0.777 - 0.628i)T \) |
| 53 | \( 1 + (-0.113 + 0.993i)T \) |
| 59 | \( 1 + (-0.957 + 0.287i)T \) |
| 61 | \( 1 + (-0.641 + 0.767i)T \) |
| 67 | \( 1 + (0.271 - 0.962i)T \) |
| 71 | \( 1 + (0.509 - 0.860i)T \) |
| 73 | \( 1 + (-0.481 + 0.876i)T \) |
| 79 | \( 1 + (-0.999 + 0.0323i)T \) |
| 83 | \( 1 + (0.271 + 0.962i)T \) |
| 89 | \( 1 + (-0.689 - 0.724i)T \) |
| 97 | \( 1 + (0.509 + 0.860i)T \) |
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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
−24.77784589317329919794581765173, −23.67144316593656513804464348695, −22.9655709625585542056243741267, −22.00253726463641681440302747762, −21.371181608850337429301696265879, −20.83892158269557781453370601111, −19.94181496590687300733909541578, −17.992742439126424116854123675656, −17.30104242325337907262427126495, −16.9559577958827554593322667527, −15.7095065927091673101512785394, −14.8298201414350254460070414545, −14.42698214872944449111744309499, −12.99987222406523792260251917059, −12.2940603077592808337971502997, −11.38660310625157584201277097153, −10.30608148186190346333856511669, −9.29787340309986799439642514744, −8.20258047398327443852636110711, −6.92438872288832434401277482227, −5.820100684468119199174727851674, −5.02886489750077718800333779695, −4.580527203210081838357074909618, −3.090083966986140558644816250408, −1.739164129240813460075200102095,
1.296672240257542011232503483411, 1.99330790423773032781525344238, 3.274585368536685048996643363115, 4.805553613241236234668065172977, 5.55288106716145770792335495516, 6.51069158098825398962850612975, 7.25746249590226204467277477976, 8.74140561330583090969714763028, 10.264270809319713410006325027700, 10.863166222007223564137537209442, 11.76853995224738636765616105087, 12.535236102872838892407246935519, 13.69751945226889217037825844810, 14.14387955991164988943056807614, 14.85955216872009545150683512398, 16.697412283686095414387819202220, 17.01789706406146950337389105781, 18.481873261888547932668048575948, 18.798564522126788715990195733501, 19.909033942720736647433919312345, 21.18344197305321436744428809309, 21.568729725293097713735351212767, 22.48534212558610367871073244010, 23.40993022098319699605014353975, 24.0758190572122838210108351552