L(s) = 1 | + (−0.0724 + 0.997i)2-s + (0.981 + 0.192i)3-s + (−0.989 − 0.144i)4-s + (−0.681 − 0.732i)5-s + (−0.262 + 0.964i)6-s + (−0.354 + 0.935i)7-s + (0.215 − 0.976i)8-s + (0.926 + 0.377i)9-s + (0.779 − 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.354 + 0.935i)13-s + (−0.906 − 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.958 + 0.285i)16-s + (−0.943 + 0.331i)17-s + (−0.443 + 0.896i)18-s + ⋯ |
L(s) = 1 | + (−0.0724 + 0.997i)2-s + (0.981 + 0.192i)3-s + (−0.989 − 0.144i)4-s + (−0.681 − 0.732i)5-s + (−0.262 + 0.964i)6-s + (−0.354 + 0.935i)7-s + (0.215 − 0.976i)8-s + (0.926 + 0.377i)9-s + (0.779 − 0.626i)10-s + (−0.943 − 0.331i)12-s + (−0.354 + 0.935i)13-s + (−0.906 − 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.958 + 0.285i)16-s + (−0.943 + 0.331i)17-s + (−0.443 + 0.896i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1872375331 + 0.3784106982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1872375331 + 0.3784106982i\) |
\(L(1)\) |
\(\approx\) |
\(0.7050058696 + 0.4992136076i\) |
\(L(1)\) |
\(\approx\) |
\(0.7050058696 + 0.4992136076i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.0724 + 0.997i)T \) |
| 3 | \( 1 + (0.981 + 0.192i)T \) |
| 5 | \( 1 + (-0.681 - 0.732i)T \) |
| 7 | \( 1 + (-0.354 + 0.935i)T \) |
| 13 | \( 1 + (-0.354 + 0.935i)T \) |
| 17 | \( 1 + (-0.943 + 0.331i)T \) |
| 19 | \( 1 + (0.0241 - 0.999i)T \) |
| 23 | \( 1 + (-0.861 + 0.506i)T \) |
| 29 | \( 1 + (-0.970 - 0.239i)T \) |
| 31 | \( 1 + (0.995 - 0.0965i)T \) |
| 37 | \( 1 + (0.885 - 0.464i)T \) |
| 41 | \( 1 + (-0.607 + 0.794i)T \) |
| 43 | \( 1 + (-0.681 - 0.732i)T \) |
| 47 | \( 1 + (0.926 + 0.377i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.943 + 0.331i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.681 + 0.732i)T \) |
| 71 | \( 1 + (0.399 + 0.916i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.906 + 0.421i)T \) |
| 83 | \( 1 + (-0.989 + 0.144i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.836 - 0.548i)T \) |
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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
−20.00066547137325868603925923820, −19.870526537719074849449256203677, −18.7709669456030479328372960039, −18.4342108125395988784819978255, −17.524795597447994089505137703642, −16.50707247581700439737058455736, −15.446452727622939866883994861613, −14.76836175361751021067706111592, −13.95257331695627389878479179317, −13.443621859688225063775576882574, −12.56655200481234243882063919817, −11.89541186041879078219159982789, −10.78410359224060766362512817070, −10.27672290391045131721752968790, −9.62373031672959908314537618280, −8.52961269628225039015570780705, −7.83571927513310336038649685165, −7.25330378350575932951764974973, −6.13682176945160630141458453144, −4.53593071962468038844362073554, −3.97799769166636618128089100071, −3.16608383535937420497137661305, −2.58399135670621894805869731993, −1.44330264097469293373048092464, −0.14265332063145642344451133366,
1.603273986885339429930607708842, 2.72244090386139913131891308906, 3.95051327525529726842908956985, 4.44384807661485391484044651272, 5.336413642659438642963988010448, 6.4362437147507538090823394432, 7.26619924020886813291658318345, 8.08702847414657297337753219551, 8.781738042256797254850505705375, 9.23679406344646235625588773964, 9.90393829758042721550202094632, 11.363830909540831463500343880370, 12.26961399896845649315572377766, 13.148904908947720000035955863844, 13.60202075109887549244734567704, 14.67435490664682143507268865037, 15.34305866003247649151298734075, 15.716250861878872468121223841746, 16.43641141058830491413325038502, 17.27009312659532223488695777454, 18.308602867060724856970912166178, 19.00181533383566350181426592968, 19.60558663029817362107339796121, 20.21715060050800360676374212958, 21.513069688881487166488578375497