L(s) = 1 | + (−0.703 + 0.710i)2-s + (0.994 + 0.104i)3-s + (−0.00951 − 0.999i)4-s + (−0.774 + 0.633i)6-s + (0.717 + 0.696i)8-s + (0.978 + 0.207i)9-s + (0.0950 − 0.995i)12-s + (−0.676 − 0.736i)13-s + (−0.999 + 0.0190i)16-s + (−0.475 − 0.879i)17-s + (−0.836 + 0.548i)18-s + (−0.991 + 0.132i)19-s + (−0.945 + 0.327i)23-s + (0.640 + 0.768i)24-s + (0.999 + 0.0380i)26-s + (0.951 + 0.309i)27-s + ⋯ |
L(s) = 1 | + (−0.703 + 0.710i)2-s + (0.994 + 0.104i)3-s + (−0.00951 − 0.999i)4-s + (−0.774 + 0.633i)6-s + (0.717 + 0.696i)8-s + (0.978 + 0.207i)9-s + (0.0950 − 0.995i)12-s + (−0.676 − 0.736i)13-s + (−0.999 + 0.0190i)16-s + (−0.475 − 0.879i)17-s + (−0.836 + 0.548i)18-s + (−0.991 + 0.132i)19-s + (−0.945 + 0.327i)23-s + (0.640 + 0.768i)24-s + (0.999 + 0.0380i)26-s + (0.951 + 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2141935797 - 0.3727025549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2141935797 - 0.3727025549i\) |
\(L(1)\) |
\(\approx\) |
\(0.8309085909 + 0.1450332959i\) |
\(L(1)\) |
\(\approx\) |
\(0.8309085909 + 0.1450332959i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.703 + 0.710i)T \) |
| 3 | \( 1 + (0.994 + 0.104i)T \) |
| 13 | \( 1 + (-0.676 - 0.736i)T \) |
| 17 | \( 1 + (-0.475 - 0.879i)T \) |
| 19 | \( 1 + (-0.991 + 0.132i)T \) |
| 23 | \( 1 + (-0.945 + 0.327i)T \) |
| 29 | \( 1 + (0.941 - 0.336i)T \) |
| 31 | \( 1 + (-0.398 - 0.917i)T \) |
| 37 | \( 1 + (0.318 - 0.948i)T \) |
| 41 | \( 1 + (0.362 + 0.931i)T \) |
| 43 | \( 1 + (-0.989 - 0.142i)T \) |
| 47 | \( 1 + (0.836 + 0.548i)T \) |
| 53 | \( 1 + (-0.0190 + 0.999i)T \) |
| 59 | \( 1 + (-0.988 + 0.151i)T \) |
| 61 | \( 1 + (-0.964 - 0.263i)T \) |
| 67 | \( 1 + (0.998 + 0.0475i)T \) |
| 71 | \( 1 + (-0.564 + 0.825i)T \) |
| 73 | \( 1 + (-0.424 - 0.905i)T \) |
| 79 | \( 1 + (-0.0665 + 0.997i)T \) |
| 83 | \( 1 + (-0.389 - 0.921i)T \) |
| 89 | \( 1 + (-0.235 + 0.971i)T \) |
| 97 | \( 1 + (-0.170 - 0.985i)T \) |
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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
−18.74277564529905546915062188918, −18.11471258454673303027275085429, −17.37036521030940070039675405438, −16.71308359913235182790331786932, −15.94403337762896054181327248568, −15.22515192384388020312810596194, −14.43879434115929566599387296637, −13.79502488955165051093970150115, −13.05479863997749681983170238416, −12.39248981839990657708307164882, −11.91580257854148357810436914615, −10.81815558091584192304039958053, −10.28668458404027002971778522119, −9.65947138160142574274487724939, −8.77543611308542120398847051927, −8.51622765073141360768475334236, −7.727992135969265620298672817656, −6.89862995612846167648790771618, −6.38893081279822070580357997769, −4.79742504346475204435243570783, −4.19314563536854657990866568969, −3.50341218719496330516163989539, −2.58551711940785357483371929467, −2.01692334534643759457362657489, −1.34819081695117171938040566908,
0.12192804551579147666278180914, 1.33376328269715936312194926383, 2.295999946087840228134104000229, 2.80620910838423677757465069787, 4.11827564547693911485044228264, 4.640814125778297549183811467122, 5.62146924158798375048137371445, 6.40721742137486779885577248351, 7.24108266485767613467093087200, 7.82975518844811156849766942629, 8.31420068863601057849945630253, 9.202819448668112749419022861583, 9.626925870600479590424192529539, 10.346880193953634691272672990808, 10.98112737841163588954227451083, 12.042643036739774592734822113, 12.91206958412555085290520955151, 13.657644370321721563414368844782, 14.22655734864949543312072630421, 14.91813411502425951867081872295, 15.434592755841210022406212711320, 16.00054766732870857274565999769, 16.763645195422227772462228870100, 17.51440989485198919436902901049, 18.18208600063369777320220493930