L(s) = 1 | + (0.978 − 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (−0.913 + 0.406i)18-s + (0.913 + 0.406i)19-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (−0.309 − 0.951i)27-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 + 0.866i)12-s + (−0.309 + 0.951i)13-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (−0.913 + 0.406i)18-s + (0.913 + 0.406i)19-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (−0.309 − 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.309089588 + 0.9392437677i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.309089588 + 0.9392437677i\) |
\(L(1)\) |
\(\approx\) |
\(1.881030745 + 0.4182945043i\) |
\(L(1)\) |
\(\approx\) |
\(1.881030745 + 0.4182945043i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.913 + 0.406i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (0.104 - 0.994i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.913 - 0.406i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 + (0.669 - 0.743i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.913 + 0.406i)T \) |
| 79 | \( 1 + (-0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−24.32979122649481327596393128697, −23.6136155023600098808162856336, −22.672345431617510715187851724899, −22.18918163478257771064029732474, −20.72383990721403662275410231673, −20.29056429835519905117241040678, −19.234049469745575598122062014838, −18.28976403956042667291063938369, −17.24467199741684604186983972336, −16.46776104982197110906523164057, −15.22049828798135391848609286555, −14.52156711090953134129077639662, −13.574244776218770421445626218585, −12.887788382765342537690259851649, −12.063409600989762707092444480353, −11.296512865095134888696374293495, −10.03030338596438316456775636589, −8.447370905186766112957766103895, −7.60154789358645925266194940931, −6.83182439217917714071558713804, −5.749944779811133557727076208057, −4.97008606613088009728944367231, −3.350227498880993662545082493086, −2.640470087656160415475105889685, −1.22497840610763525889991486584,
1.70821648221569449127364499629, 3.10851950338303612033231580022, 3.82090974117708174111836359549, 4.942138716137916749734779952120, 5.6280641851661504463308620056, 6.87418155452804055025457452306, 8.07595033996045048760085205645, 9.54281296300669948411452129184, 10.09978915762941839069482095640, 11.36150735035705836766961782389, 11.82097720916488557681590880584, 13.11013308019631382054222742997, 14.16917677674629153632850935913, 14.65252543324152620991597485417, 15.686758488618046210880542945192, 16.38504810506003745138982305052, 17.20181212591982374111810205691, 18.814042187851378437897972801946, 19.65840224163009644100438348144, 20.54366771056582223845111203338, 21.337171949591172994570780693986, 21.80086039426666165755592733455, 22.86866963085212395141887033348, 23.40731557731552501555334941511, 24.56824082062389581813862345062