L(s) = 1 | + (0.953 + 0.301i)2-s + (−0.771 − 0.636i)3-s + (0.817 + 0.575i)4-s + (0.896 + 0.443i)5-s + (−0.543 − 0.839i)6-s + (0.720 + 0.693i)7-s + (0.606 + 0.795i)8-s + (0.190 + 0.981i)9-s + (0.720 + 0.693i)10-s + (−0.927 + 0.373i)11-s + (−0.264 − 0.964i)12-s + (0.953 + 0.301i)13-s + (0.477 + 0.878i)14-s + (−0.409 − 0.912i)15-s + (0.338 + 0.941i)16-s + (−0.771 + 0.636i)17-s + ⋯ |
L(s) = 1 | + (0.953 + 0.301i)2-s + (−0.771 − 0.636i)3-s + (0.817 + 0.575i)4-s + (0.896 + 0.443i)5-s + (−0.543 − 0.839i)6-s + (0.720 + 0.693i)7-s + (0.606 + 0.795i)8-s + (0.190 + 0.981i)9-s + (0.720 + 0.693i)10-s + (−0.927 + 0.373i)11-s + (−0.264 − 0.964i)12-s + (0.953 + 0.301i)13-s + (0.477 + 0.878i)14-s + (−0.409 − 0.912i)15-s + (0.338 + 0.941i)16-s + (−0.771 + 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.200423995 + 1.877739987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.200423995 + 1.877739987i\) |
\(L(1)\) |
\(\approx\) |
\(1.730940166 + 0.5988062016i\) |
\(L(1)\) |
\(\approx\) |
\(1.730940166 + 0.5988062016i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + (0.953 + 0.301i)T \) |
| 3 | \( 1 + (-0.771 - 0.636i)T \) |
| 5 | \( 1 + (0.896 + 0.443i)T \) |
| 7 | \( 1 + (0.720 + 0.693i)T \) |
| 11 | \( 1 + (-0.927 + 0.373i)T \) |
| 13 | \( 1 + (0.953 + 0.301i)T \) |
| 17 | \( 1 + (-0.771 + 0.636i)T \) |
| 19 | \( 1 + (-0.665 - 0.746i)T \) |
| 23 | \( 1 + (0.190 - 0.981i)T \) |
| 29 | \( 1 + (0.988 - 0.152i)T \) |
| 31 | \( 1 + (0.0383 + 0.999i)T \) |
| 37 | \( 1 + (0.817 - 0.575i)T \) |
| 43 | \( 1 + (-0.927 + 0.373i)T \) |
| 47 | \( 1 + (0.720 + 0.693i)T \) |
| 53 | \( 1 + (-0.927 + 0.373i)T \) |
| 59 | \( 1 + (0.0383 - 0.999i)T \) |
| 61 | \( 1 + (-0.771 - 0.636i)T \) |
| 67 | \( 1 + (-0.997 - 0.0765i)T \) |
| 71 | \( 1 + (0.988 + 0.152i)T \) |
| 73 | \( 1 + (-0.114 + 0.993i)T \) |
| 79 | \( 1 + (0.0383 - 0.999i)T \) |
| 83 | \( 1 + (-0.927 - 0.373i)T \) |
| 89 | \( 1 + (-0.997 + 0.0765i)T \) |
| 97 | \( 1 + (0.606 - 0.795i)T \) |
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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
−20.59757970225541141712664768155, −19.90269128866860735945331197200, −18.46226770221293883297082061301, −17.99564377401876502056517426271, −16.992923627101117952410587073713, −16.492263887800656930218795095575, −15.633705049316739179344416532740, −15.056161116705376336142120393476, −13.96224436451685221698853123997, −13.45591167773322946108438941234, −12.85516877521384920776173622839, −11.78046614458375789414057340051, −11.14879746553393002040087699458, −10.458754675883201682995776455824, −10.017631847980387014201485021328, −8.88725217053783771188460004161, −7.820311417541897855799129485025, −6.64347611008819788024774828015, −5.947347824972603762018346794006, −5.28745138851736352301311612800, −4.637378764601174262588397568771, −3.90430929926967759257621607417, −2.840757418355041239374349263329, −1.68412540028670547747161955581, −0.81412763026561224288125991284,
1.5195965528927003037100248976, 2.2199566656763253468610080989, 2.873396373825008127259810798592, 4.548270526863826790485284196078, 4.92327439916229471031643645417, 5.99280567802380737809589205755, 6.308724606866337779652178526481, 7.09921070789243118305634998035, 8.12578712809297895720496208086, 8.79745591285466570161687675863, 10.39277328775652321597987062181, 10.92435617511754109480314321251, 11.45383133361225447068281507837, 12.72462408354298556865913895479, 12.78946673747027796340659528588, 13.79017586871112917302664469304, 14.363637687221641938586141246031, 15.35836148177514631182431996585, 15.86715669180409942898931753217, 16.92657255950312712499166277535, 17.59783480453386798866475753307, 18.11098786029329902319000857014, 18.7807122956725485150472531872, 19.88719833352257626080503624453, 20.94926351778372049572108946745