L(s) = 1 | + (−0.0275 − 0.999i)2-s + (−0.998 + 0.0550i)4-s + (0.451 − 0.892i)5-s + (0.0825 + 0.996i)8-s + (−0.904 − 0.426i)10-s + (0.298 + 0.954i)11-s + (0.986 − 0.164i)13-s + (0.993 − 0.110i)16-s + (0.716 + 0.697i)17-s + (−0.904 + 0.426i)19-s + (−0.401 + 0.915i)20-s + (0.945 − 0.324i)22-s + (−0.904 + 0.426i)23-s + (−0.592 − 0.805i)25-s + (−0.191 − 0.981i)26-s + ⋯ |
L(s) = 1 | + (−0.0275 − 0.999i)2-s + (−0.998 + 0.0550i)4-s + (0.451 − 0.892i)5-s + (0.0825 + 0.996i)8-s + (−0.904 − 0.426i)10-s + (0.298 + 0.954i)11-s + (0.986 − 0.164i)13-s + (0.993 − 0.110i)16-s + (0.716 + 0.697i)17-s + (−0.904 + 0.426i)19-s + (−0.401 + 0.915i)20-s + (0.945 − 0.324i)22-s + (−0.904 + 0.426i)23-s + (−0.592 − 0.805i)25-s + (−0.191 − 0.981i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.405908284 - 0.03671422600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405908284 - 0.03671422600i\) |
\(L(1)\) |
\(\approx\) |
\(0.9433572432 - 0.4096941188i\) |
\(L(1)\) |
\(\approx\) |
\(0.9433572432 - 0.4096941188i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (-0.0275 - 0.999i)T \) |
| 5 | \( 1 + (0.451 - 0.892i)T \) |
| 11 | \( 1 + (0.298 + 0.954i)T \) |
| 13 | \( 1 + (0.986 - 0.164i)T \) |
| 17 | \( 1 + (0.716 + 0.697i)T \) |
| 19 | \( 1 + (-0.904 + 0.426i)T \) |
| 23 | \( 1 + (-0.904 + 0.426i)T \) |
| 29 | \( 1 + (0.401 + 0.915i)T \) |
| 31 | \( 1 + (0.754 - 0.656i)T \) |
| 37 | \( 1 + (-0.754 - 0.656i)T \) |
| 41 | \( 1 + (-0.879 + 0.475i)T \) |
| 43 | \( 1 + (0.789 + 0.614i)T \) |
| 47 | \( 1 + (0.975 + 0.218i)T \) |
| 53 | \( 1 + (0.298 + 0.954i)T \) |
| 59 | \( 1 + (-0.754 + 0.656i)T \) |
| 61 | \( 1 + (-0.716 + 0.697i)T \) |
| 67 | \( 1 + (-0.962 - 0.272i)T \) |
| 71 | \( 1 + (0.879 - 0.475i)T \) |
| 73 | \( 1 + (-0.975 + 0.218i)T \) |
| 79 | \( 1 + (0.993 - 0.110i)T \) |
| 83 | \( 1 + (-0.0825 - 0.996i)T \) |
| 89 | \( 1 + (-0.926 - 0.376i)T \) |
| 97 | \( 1 + (-0.945 + 0.324i)T \) |
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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
−18.40719672769885921268054555248, −17.77056748260596906829147013520, −17.05824977486494996989537209103, −16.48984324499713974589420962481, −15.61147846328883286825792211118, −15.30142638438980850745819046149, −14.16451697177367891616755763180, −13.94172246707268587631864113853, −13.49165394976726711716395309931, −12.40180590620096911525148957185, −11.60523399023559764986476321630, −10.6880422347776779667559442015, −10.18184930104045139676821272107, −9.34700315131779928756767952245, −8.55768423567576788256151337465, −8.09134947733088543133133148669, −7.05568159795611129455936144345, −6.49703143418025442479502421154, −5.98636809276923831504502729946, −5.28813357558473946109100449600, −4.23024043205485248504455042671, −3.537109862274722383625393091242, −2.75553994157812303372628279153, −1.56719402802887591412748770659, −0.43607181535624922663297334874,
1.10179364932008206429784311022, 1.58316008411536654400364725485, 2.346017137520447524042486842517, 3.460741934269958844988014018558, 4.18533973871794637033610718530, 4.698092169990887697880906715210, 5.75638503958035976695381593093, 6.14375022367978387352789280652, 7.550033241678532020078392850391, 8.27117132405826248580170919186, 8.89022655833977075077385076613, 9.51237034827412103153749776732, 10.34420211567888929582786966600, 10.64704751081509307168846397502, 11.88689451018523361945484937344, 12.2510899421498094505752732651, 12.84198268249654440821664138552, 13.54212447840822953182097780125, 14.14214766532024246960665018879, 14.94242829653701482177511873442, 15.784157475642303790126655864968, 16.74232956671257757956754602499, 17.186394155909939244022488107301, 17.901667329951062936125304740267, 18.43075754166803042440958331269