L(s) = 1 | + (0.120 − 0.992i)2-s + (0.995 − 0.0965i)3-s + (−0.970 − 0.239i)4-s + (0.399 + 0.916i)5-s + (0.0241 − 0.999i)6-s + (0.943 − 0.331i)7-s + (−0.354 + 0.935i)8-s + (0.981 − 0.192i)9-s + (0.958 − 0.285i)10-s + (−0.989 − 0.144i)12-s + (0.607 + 0.794i)13-s + (−0.215 − 0.976i)14-s + (0.485 + 0.873i)15-s + (0.885 + 0.464i)16-s + (−0.443 − 0.896i)17-s + (−0.0724 − 0.997i)18-s + ⋯ |
L(s) = 1 | + (0.120 − 0.992i)2-s + (0.995 − 0.0965i)3-s + (−0.970 − 0.239i)4-s + (0.399 + 0.916i)5-s + (0.0241 − 0.999i)6-s + (0.943 − 0.331i)7-s + (−0.354 + 0.935i)8-s + (0.981 − 0.192i)9-s + (0.958 − 0.285i)10-s + (−0.989 − 0.144i)12-s + (0.607 + 0.794i)13-s + (−0.215 − 0.976i)14-s + (0.485 + 0.873i)15-s + (0.885 + 0.464i)16-s + (−0.443 − 0.896i)17-s + (−0.0724 − 0.997i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.359186741 - 1.419979284i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.359186741 - 1.419979284i\) |
\(L(1)\) |
\(\approx\) |
\(1.582538237 - 0.7022675948i\) |
\(L(1)\) |
\(\approx\) |
\(1.582538237 - 0.7022675948i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (0.120 - 0.992i)T \) |
| 3 | \( 1 + (0.995 - 0.0965i)T \) |
| 5 | \( 1 + (0.399 + 0.916i)T \) |
| 7 | \( 1 + (0.943 - 0.331i)T \) |
| 13 | \( 1 + (0.607 + 0.794i)T \) |
| 17 | \( 1 + (-0.443 - 0.896i)T \) |
| 19 | \( 1 + (-0.168 - 0.985i)T \) |
| 23 | \( 1 + (-0.779 + 0.626i)T \) |
| 29 | \( 1 + (0.485 - 0.873i)T \) |
| 31 | \( 1 + (0.998 + 0.0483i)T \) |
| 37 | \( 1 + (-0.644 - 0.764i)T \) |
| 41 | \( 1 + (-0.715 - 0.698i)T \) |
| 43 | \( 1 + (-0.995 + 0.0965i)T \) |
| 47 | \( 1 + (0.906 + 0.421i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.885 + 0.464i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.995 - 0.0965i)T \) |
| 71 | \( 1 + (0.262 + 0.964i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.995 - 0.0965i)T \) |
| 83 | \( 1 + (-0.0724 + 0.997i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (-0.958 - 0.285i)T \) |
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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
−20.81565094071596973250742772653, −20.333844967578159160219661998, −19.31655694904506815364727137061, −18.315050617195082875964256376288, −17.89717474253507061458042508235, −16.956343499429241532747126051709, −16.25881458157348384869774136628, −15.363858834527087730006268952938, −14.95756718721313402826165965164, −14.00588080365803680449543658002, −13.55525208256672097692301543361, −12.683615048523082051326885800404, −12.09357109750914619499572885496, −10.451803643132851880603615094613, −9.88525466332205841383923695751, −8.67976553238719884048929272758, −8.38401634962520912975033577621, −8.02164929478988523550058381956, −6.71601148668503629252950878202, −5.819566145392608433250278909744, −4.97848121816928896236020206072, −4.27962622090047325706609697305, −3.4131148429142096615189756137, −2.03600589177158208180179864400, −1.163984469934560537252430894347,
1.102643008306919706909566084697, 2.15021966671199661900295226563, 2.523499695102220981494934614955, 3.68481531740533509834706081277, 4.27587577334076472432924438723, 5.28027527425359373575728747632, 6.58452783732289073308527886758, 7.38050898274361117738766653481, 8.378801930751287195121726533206, 9.04001276374669963262553562654, 9.88623926390691227684815523917, 10.54985044115467396939756994113, 11.4255999264468586430004756709, 11.93432551747767399757977867561, 13.38091951892464219511790859282, 13.74222028837020973899296115658, 14.13449824824789163705700003502, 15.04855477697679456507934888021, 15.72686557361711053114773201791, 17.2468012567834274122119835549, 17.96835112784274268858012685927, 18.39786616467824093774296622166, 19.280546218122453775111058810786, 19.78098904539513061545406628570, 20.7864756351333301801801817401