L(s) = 1 | + (0.173 − 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (0.866 − 0.5i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (−0.939 + 0.342i)14-s − i·15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (0.866 − 0.5i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (−0.939 + 0.342i)14-s − i·15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-1.201797488 - 1.835264155i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-1.201797488 - 1.835264155i\) |
\(L(1)\) |
\(\approx\) |
\(0.5224647585 - 1.263253291i\) |
\(L(1)\) |
\(\approx\) |
\(0.5224647585 - 1.263253291i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.342 - 0.939i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.642 + 0.766i)T \) |
| 31 | \( 1 + (0.342 - 0.939i)T \) |
| 41 | \( 1 + (-0.642 - 0.766i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.342 - 0.939i)T \) |
| 53 | \( 1 + (-0.984 + 0.173i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + (0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.766 + 0.642i)T \) |
| 89 | \( 1 + (-0.642 - 0.766i)T \) |
| 97 | \( 1 + (-0.642 + 0.766i)T \) |
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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
−18.85866358288110188153873809565, −18.09736310771417188815355259317, −17.28004634503356495104027330503, −17.02089102141492817607424997721, −15.94022636868659483158184931108, −15.51048460154471408896597696597, −14.85812077679078336414958043615, −14.41727668927221766815468069893, −13.80444624016815472607358004813, −12.86182639653228288861364052961, −12.42790040664121210083525119473, −11.34770079456904315701776308368, −10.117674604444591783506768595651, −9.76825607935126591024793592309, −9.44036829020558988200717921383, −8.493003763738392861567977771906, −7.75221485506351123312240752177, −7.05963664978411861954773546516, −6.1022692371056874543017333552, −5.40754878567536305404730921287, −5.15022654717960607819389858679, −3.948102240300652018852512847560, −3.208815904326198907366157324579, −2.65429163070816985479080416526, −1.49023316155127231843778915927,
0.59052500562118616385907973905, 1.078124126625894864363243779242, 1.9542111626870687550386063074, 2.781886827426288945993854424546, 3.39564842044323775617621340183, 4.25953394674629224746323217417, 5.20001071631421679782513995339, 5.953400616021632709615023821393, 6.73844078655530567463333556290, 7.5092168956122228020322280049, 8.50835553582425918839893747595, 9.17490557824302220302581259765, 9.56266873682250095169517537834, 10.37176309903003677494607693151, 11.25742250978221740526006794442, 11.94964097299602749172113629846, 12.62139079662982840606183064256, 13.26219181899544398711631992606, 13.781538510611355467582839964009, 14.168662677873432217148066320926, 14.79469626381313861572465593397, 16.258040077339127622410999716696, 16.88791883120586303164189239931, 17.35932379810735542786466104226, 18.28497635419137732776396499429