L(s) = 1 | + (−0.201 − 0.979i)2-s + (0.445 + 0.895i)3-s + (−0.918 + 0.395i)4-s + (0.786 − 0.617i)5-s + (0.786 − 0.617i)6-s + (−0.542 − 0.840i)7-s + (0.572 + 0.819i)8-s + (−0.602 + 0.798i)9-s + (−0.763 − 0.645i)10-s + (0.237 − 0.971i)11-s + (−0.763 − 0.645i)12-s + (0.739 − 0.673i)13-s + (−0.713 + 0.700i)14-s + (0.903 + 0.429i)15-s + (0.687 − 0.726i)16-s + (−0.999 + 0.0369i)17-s + ⋯ |
L(s) = 1 | + (−0.201 − 0.979i)2-s + (0.445 + 0.895i)3-s + (−0.918 + 0.395i)4-s + (0.786 − 0.617i)5-s + (0.786 − 0.617i)6-s + (−0.542 − 0.840i)7-s + (0.572 + 0.819i)8-s + (−0.602 + 0.798i)9-s + (−0.763 − 0.645i)10-s + (0.237 − 0.971i)11-s + (−0.763 − 0.645i)12-s + (0.739 − 0.673i)13-s + (−0.713 + 0.700i)14-s + (0.903 + 0.429i)15-s + (0.687 − 0.726i)16-s + (−0.999 + 0.0369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3719668510 - 1.098537905i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3719668510 - 1.098537905i\) |
\(L(1)\) |
\(\approx\) |
\(0.8598185285 - 0.5076109848i\) |
\(L(1)\) |
\(\approx\) |
\(0.8598185285 - 0.5076109848i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.201 - 0.979i)T \) |
| 3 | \( 1 + (0.445 + 0.895i)T \) |
| 5 | \( 1 + (0.786 - 0.617i)T \) |
| 7 | \( 1 + (-0.542 - 0.840i)T \) |
| 11 | \( 1 + (0.237 - 0.971i)T \) |
| 13 | \( 1 + (0.739 - 0.673i)T \) |
| 17 | \( 1 + (-0.999 + 0.0369i)T \) |
| 19 | \( 1 + (-0.602 - 0.798i)T \) |
| 23 | \( 1 + (0.687 + 0.726i)T \) |
| 29 | \( 1 + (-0.542 + 0.840i)T \) |
| 31 | \( 1 + (0.0184 - 0.999i)T \) |
| 37 | \( 1 + (0.989 - 0.147i)T \) |
| 41 | \( 1 + (-0.602 - 0.798i)T \) |
| 43 | \( 1 + (-0.966 + 0.255i)T \) |
| 47 | \( 1 + (-0.713 + 0.700i)T \) |
| 53 | \( 1 + (-0.128 - 0.991i)T \) |
| 59 | \( 1 + (-0.886 - 0.462i)T \) |
| 61 | \( 1 + (-0.713 + 0.700i)T \) |
| 67 | \( 1 + (0.309 - 0.951i)T \) |
| 71 | \( 1 + (0.572 + 0.819i)T \) |
| 73 | \( 1 + (-0.713 - 0.700i)T \) |
| 79 | \( 1 + (0.997 + 0.0738i)T \) |
| 83 | \( 1 + (0.445 + 0.895i)T \) |
| 89 | \( 1 + (0.0922 - 0.995i)T \) |
| 97 | \( 1 + (0.932 + 0.361i)T \) |
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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
−22.096292716050682468839080493945, −21.28848528997958504368367494094, −20.14395749039941570732210878982, −19.20925609271883935796161352526, −18.42246793925000794293508477231, −18.28609542523563133597185956892, −17.29099754997310768933127145508, −16.583608314151166645235727894784, −15.3010275612830065886667952190, −14.92151793606251062994214986955, −14.117669534575676620642007894778, −13.27096420750340367881762338071, −12.80998551918663545141057434582, −11.71367712701022295640967483455, −10.4429543552064252630858383765, −9.440345968513240704911354263746, −8.97016027919355095611378970211, −8.1141668694327987737842289938, −6.95746799891109467800539579375, −6.47327499247098464664897765013, −6.00776316259991145825908308699, −4.73595636136450941358635923781, −3.478829971468620786627290258762, −2.280578274415811227971336015861, −1.53575427541209636456811164508,
0.4873200144803015555327339599, 1.715124898050008163686641713028, 2.896966988142301392735554811706, 3.58870130489257678466737894062, 4.45922364305554849889682460865, 5.30345939263302321837796293440, 6.36218026673053790344310452673, 7.89617506358243684759714442490, 8.78554064839880225600276539098, 9.240759030010872393213202403351, 10.02341153815866505587843622220, 10.93674906193409929861166543381, 11.202566323576278162384263492968, 12.8578435899695746881341675781, 13.39296582617588240837980172840, 13.73868101874464821842219062303, 14.89113825319418897016047433149, 15.983377934582093864513885550339, 16.78700502378272455648347284738, 17.25039710353134335533244443961, 18.2191421578572939538235539918, 19.34121863993123769253006210994, 19.91702419570948417814495924034, 20.49384777415969500968132241774, 21.186164678331772402778552412325