L(s) = 1 | + (0.869 − 0.493i)2-s + (0.823 + 0.566i)3-s + (0.512 − 0.858i)4-s + (0.192 − 0.981i)5-s + (0.996 + 0.0859i)6-s + (−0.999 + 0.0430i)7-s + (0.0215 − 0.999i)8-s + (0.357 + 0.933i)9-s + (−0.317 − 0.948i)10-s + (−0.954 − 0.296i)11-s + (0.908 − 0.417i)12-s + (0.985 + 0.171i)13-s + (−0.847 + 0.530i)14-s + (0.714 − 0.699i)15-s + (−0.474 − 0.880i)16-s + (0.869 − 0.493i)17-s + ⋯ |
L(s) = 1 | + (0.869 − 0.493i)2-s + (0.823 + 0.566i)3-s + (0.512 − 0.858i)4-s + (0.192 − 0.981i)5-s + (0.996 + 0.0859i)6-s + (−0.999 + 0.0430i)7-s + (0.0215 − 0.999i)8-s + (0.357 + 0.933i)9-s + (−0.317 − 0.948i)10-s + (−0.954 − 0.296i)11-s + (0.908 − 0.417i)12-s + (0.985 + 0.171i)13-s + (−0.847 + 0.530i)14-s + (0.714 − 0.699i)15-s + (−0.474 − 0.880i)16-s + (0.869 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.156356994 - 1.287762698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.156356994 - 1.287762698i\) |
\(L(1)\) |
\(\approx\) |
\(1.896955389 - 0.6826513252i\) |
\(L(1)\) |
\(\approx\) |
\(1.896955389 - 0.6826513252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 293 | \( 1 \) |
good | 2 | \( 1 + (0.869 - 0.493i)T \) |
| 3 | \( 1 + (0.823 + 0.566i)T \) |
| 5 | \( 1 + (0.192 - 0.981i)T \) |
| 7 | \( 1 + (-0.999 + 0.0430i)T \) |
| 11 | \( 1 + (-0.954 - 0.296i)T \) |
| 13 | \( 1 + (0.985 + 0.171i)T \) |
| 17 | \( 1 + (0.869 - 0.493i)T \) |
| 19 | \( 1 + (0.714 - 0.699i)T \) |
| 23 | \( 1 + (0.996 - 0.0859i)T \) |
| 29 | \( 1 + (-0.798 + 0.601i)T \) |
| 31 | \( 1 + (0.192 + 0.981i)T \) |
| 37 | \( 1 + (-0.548 + 0.835i)T \) |
| 41 | \( 1 + (-0.991 - 0.128i)T \) |
| 43 | \( 1 + (-0.0645 + 0.997i)T \) |
| 47 | \( 1 + (-0.976 - 0.213i)T \) |
| 53 | \( 1 + (-0.954 - 0.296i)T \) |
| 59 | \( 1 + (0.651 + 0.758i)T \) |
| 61 | \( 1 + (0.584 + 0.811i)T \) |
| 67 | \( 1 + (-0.150 - 0.988i)T \) |
| 71 | \( 1 + (-0.618 + 0.785i)T \) |
| 73 | \( 1 + (0.512 + 0.858i)T \) |
| 79 | \( 1 + (0.584 + 0.811i)T \) |
| 83 | \( 1 + (-0.234 + 0.972i)T \) |
| 89 | \( 1 + (0.584 - 0.811i)T \) |
| 97 | \( 1 + (0.436 - 0.899i)T \) |
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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
−25.573673532666844508528851069520, −24.92319499197549201160677805440, −23.57456674871103127103581712429, −23.103173586206168170616830487750, −22.24294368355661260256185708688, −21.0077954771365757832056170823, −20.51561437363729178401320601432, −19.07883654559078301392281133363, −18.56508207480653469335930843096, −17.40625899985605892305097292309, −16.07749008502979194854615612843, −15.287672084612656191601765809282, −14.538164571353492511821786913004, −13.47114660946610026419046785293, −13.09172034434514159942397945112, −11.991304671581668056279818765687, −10.64923075174514278447573535258, −9.54276709804249525963600311636, −8.08705612681154772119732173256, −7.36895919785549753530409342606, −6.43051656062811932963676477680, −5.60719471104175878257047499430, −3.611915503046440578408370287855, −3.234741463058664839106172306298, −2.049936680130581279151151924151,
1.31784106411629252451381838520, 2.8934960368183088820094640226, 3.48692187131337010818893926851, 4.86058030234919709465695511038, 5.5155184156866021509529753771, 6.992683028163176921101414466235, 8.4633529884706706480990656145, 9.433956400157975860571982186273, 10.17198409642124371107156774432, 11.30623686200044784615132576001, 12.653797731749479817001878222146, 13.31657949760443784048081530585, 13.87546085769251691659497325679, 15.21690730505977977520614600650, 16.05187252620963074127287529420, 16.440899245425375645149061012512, 18.44543987336153294806340119799, 19.29260637376416654713371604120, 20.186299344728031541224730415867, 20.87058799960808550395059054285, 21.41539569744143906593307915224, 22.48558444110104894963318661021, 23.42991170992834360316190075836, 24.34587135785502712060782953480, 25.33609012812549907068976008071