L(s) = 1 | + (−0.412 − 0.911i)2-s + (0.445 − 0.895i)3-s + (−0.659 + 0.751i)4-s + (−0.999 − 0.0369i)5-s + (−0.999 − 0.0369i)6-s + (−0.0554 − 0.998i)7-s + (0.956 + 0.291i)8-s + (−0.602 − 0.798i)9-s + (0.378 + 0.925i)10-s + (0.997 + 0.0738i)11-s + (0.378 + 0.925i)12-s + (0.739 + 0.673i)13-s + (−0.886 + 0.462i)14-s + (−0.478 + 0.878i)15-s + (−0.128 − 0.991i)16-s + (−0.343 + 0.938i)17-s + ⋯ |
L(s) = 1 | + (−0.412 − 0.911i)2-s + (0.445 − 0.895i)3-s + (−0.659 + 0.751i)4-s + (−0.999 − 0.0369i)5-s + (−0.999 − 0.0369i)6-s + (−0.0554 − 0.998i)7-s + (0.956 + 0.291i)8-s + (−0.602 − 0.798i)9-s + (0.378 + 0.925i)10-s + (0.997 + 0.0738i)11-s + (0.378 + 0.925i)12-s + (0.739 + 0.673i)13-s + (−0.886 + 0.462i)14-s + (−0.478 + 0.878i)15-s + (−0.128 − 0.991i)16-s + (−0.343 + 0.938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8338129818 - 0.2523838531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8338129818 - 0.2523838531i\) |
\(L(1)\) |
\(\approx\) |
\(0.6716775749 - 0.4164441638i\) |
\(L(1)\) |
\(\approx\) |
\(0.6716775749 - 0.4164441638i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.412 - 0.911i)T \) |
| 3 | \( 1 + (0.445 - 0.895i)T \) |
| 5 | \( 1 + (-0.999 - 0.0369i)T \) |
| 7 | \( 1 + (-0.0554 - 0.998i)T \) |
| 11 | \( 1 + (0.997 + 0.0738i)T \) |
| 13 | \( 1 + (0.739 + 0.673i)T \) |
| 17 | \( 1 + (-0.343 + 0.938i)T \) |
| 19 | \( 1 + (-0.602 + 0.798i)T \) |
| 23 | \( 1 + (-0.128 + 0.991i)T \) |
| 29 | \( 1 + (-0.0554 + 0.998i)T \) |
| 31 | \( 1 + (0.572 - 0.819i)T \) |
| 37 | \( 1 + (0.165 + 0.986i)T \) |
| 41 | \( 1 + (-0.602 + 0.798i)T \) |
| 43 | \( 1 + (0.631 + 0.775i)T \) |
| 47 | \( 1 + (-0.886 + 0.462i)T \) |
| 53 | \( 1 + (0.903 + 0.429i)T \) |
| 59 | \( 1 + (0.989 + 0.147i)T \) |
| 61 | \( 1 + (-0.886 + 0.462i)T \) |
| 67 | \( 1 + (-0.809 + 0.587i)T \) |
| 71 | \( 1 + (0.956 + 0.291i)T \) |
| 73 | \( 1 + (-0.886 - 0.462i)T \) |
| 79 | \( 1 + (-0.763 + 0.645i)T \) |
| 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
−21.88645398109524887294416013097, −20.81649814238577846117053301187, −19.87873783147964941741887112573, −19.36292224666707548931119130856, −18.58373446080008660729892617116, −17.68992834794238016913785550662, −16.7210972094733754209360791131, −15.89657157111214563790969227247, −15.563196542695770417651630314137, −14.882507708761512889167472947783, −14.18670213696434694777572749812, −13.18767198978686573675918513788, −11.97087333139227586386041586870, −11.12479684576213413093278478354, −10.31208241327160957011713904866, −9.17864501934940368916449575001, −8.73332211191318421564174163570, −8.17390156477124880562886005578, −7.04338179738133832662327453860, −6.14339355808368681942305128734, −5.11430626170110149445899742379, −4.36221368912139204277616526292, −3.48819171689372816922721568057, −2.31296638028646150528187231456, −0.47285857224140128255418628572,
1.16823954540136066310218921868, 1.60168171729647672517260227015, 3.11351683762612511955977337661, 3.87111694634618200280829610018, 4.33478666257074703901035242778, 6.26975313165334298051305818140, 7.06706917012410267735790616307, 7.93508089704298516552246377067, 8.49579011289939700248181839384, 9.33817992656486172594723843391, 10.42220263130242901633709091055, 11.40936371406869015002563151189, 11.77161676232505625542144794118, 12.78268226679564800566362675732, 13.36264325874465176212138615219, 14.23067832340099908129925203575, 14.97878387487787558565575048915, 16.40869308680128061023772281936, 16.97339928954737887039583214284, 17.80048848451587785313787822576, 18.73019239276765538426464218377, 19.37144343667721138150030296413, 19.76882443569245220236388503465, 20.440569112549225700918329030699, 21.22560934327880135514586579358