L(s) = 1 | + (0.237 + 0.971i)2-s + (0.739 + 0.673i)3-s + (−0.886 + 0.462i)4-s + (−0.478 + 0.878i)5-s + (−0.478 + 0.878i)6-s + (−0.999 − 0.0369i)7-s + (−0.659 − 0.751i)8-s + (0.0922 + 0.995i)9-s + (−0.966 − 0.255i)10-s + (−0.542 − 0.840i)11-s + (−0.966 − 0.255i)12-s + (−0.850 − 0.526i)13-s + (−0.201 − 0.979i)14-s + (−0.945 + 0.326i)15-s + (0.572 − 0.819i)16-s + (0.687 + 0.726i)17-s + ⋯ |
L(s) = 1 | + (0.237 + 0.971i)2-s + (0.739 + 0.673i)3-s + (−0.886 + 0.462i)4-s + (−0.478 + 0.878i)5-s + (−0.478 + 0.878i)6-s + (−0.999 − 0.0369i)7-s + (−0.659 − 0.751i)8-s + (0.0922 + 0.995i)9-s + (−0.966 − 0.255i)10-s + (−0.542 − 0.840i)11-s + (−0.966 − 0.255i)12-s + (−0.850 − 0.526i)13-s + (−0.201 − 0.979i)14-s + (−0.945 + 0.326i)15-s + (0.572 − 0.819i)16-s + (0.687 + 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1480844830 - 0.03818692090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1480844830 - 0.03818692090i\) |
\(L(1)\) |
\(\approx\) |
\(0.5791893963 + 0.5986278037i\) |
\(L(1)\) |
\(\approx\) |
\(0.5791893963 + 0.5986278037i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (0.237 + 0.971i)T \) |
| 3 | \( 1 + (0.739 + 0.673i)T \) |
| 5 | \( 1 + (-0.478 + 0.878i)T \) |
| 7 | \( 1 + (-0.999 - 0.0369i)T \) |
| 11 | \( 1 + (-0.542 - 0.840i)T \) |
| 13 | \( 1 + (-0.850 - 0.526i)T \) |
| 17 | \( 1 + (0.687 + 0.726i)T \) |
| 19 | \( 1 + (0.0922 - 0.995i)T \) |
| 23 | \( 1 + (0.572 + 0.819i)T \) |
| 29 | \( 1 + (-0.999 + 0.0369i)T \) |
| 31 | \( 1 + (-0.918 + 0.395i)T \) |
| 37 | \( 1 + (-0.993 - 0.110i)T \) |
| 41 | \( 1 + (0.0922 - 0.995i)T \) |
| 43 | \( 1 + (0.830 - 0.557i)T \) |
| 47 | \( 1 + (-0.201 - 0.979i)T \) |
| 53 | \( 1 + (0.956 - 0.291i)T \) |
| 59 | \( 1 + (-0.412 + 0.911i)T \) |
| 61 | \( 1 + (-0.201 - 0.979i)T \) |
| 67 | \( 1 + (-0.809 + 0.587i)T \) |
| 71 | \( 1 + (-0.659 - 0.751i)T \) |
| 73 | \( 1 + (-0.201 + 0.979i)T \) |
| 79 | \( 1 + (-0.0554 - 0.998i)T \) |
| 83 | \( 1 + (0.739 + 0.673i)T \) |
| 89 | \( 1 + (0.445 + 0.895i)T \) |
| 97 | \( 1 + (-0.273 - 0.961i)T \) |
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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
−21.33497733421633989971819529196, −20.57213165126606797594448920269, −20.25647050664430908433145728711, −19.33619038181089507961382617074, −18.902670312000611111675543528960, −18.16817508147408047432593554096, −17.02587945256755389504416128489, −16.21109275037959839808081702569, −15.05163983571492510466047297340, −14.45310100579422822446978998506, −13.43038346534062126500073333102, −12.71864896111359242249852386878, −12.392319963490638098145308988504, −11.683966286926266727373702434081, −10.27939693641714879074586207740, −9.43301855455422425279830175736, −9.10395304094016208313227133425, −7.88543008055983105311848317577, −7.24165328333097672634055183361, −5.92263053268278184908595007562, −4.88418676459769637079364624437, −3.98016398475187404182142660499, −3.10199766869741488608791599043, −2.24049083188594880603555223157, −1.24418811897501635113814838757,
0.05562684525111817009255484119, 2.58849700160892322056776833067, 3.404937398361053993035500934198, 3.77382930536445602493921479299, 5.17577454769849665060779947628, 5.815775007358438785984305458515, 7.19273385005840962312673674383, 7.43196939592500137134979320093, 8.54898665210937574874053555638, 9.27610222320517129874921180060, 10.19597532265197630952043330260, 10.84442718201336425870798883025, 12.22479224758669422608028506230, 13.19745059107234305296834521339, 13.76147630184474288818666142317, 14.730377186754728626549416084456, 15.23261338819968056331308880127, 15.84109069359797374291557815404, 16.552128963833344357095904519540, 17.40504594869519064526900447591, 18.56834333082755278816814932610, 19.22623219279859425298578502569, 19.70005342455164826578063110464, 20.99905876727585933431907664104, 21.95222083998886339709466120989