L(s) = 1 | + (−0.251 + 0.967i)2-s + (−0.873 − 0.486i)4-s + (0.791 + 0.611i)5-s + (0.691 − 0.722i)8-s + (−0.791 + 0.611i)10-s + (−0.447 + 0.894i)11-s + (−0.963 + 0.266i)13-s + (0.525 + 0.850i)16-s + (−0.0149 + 0.999i)17-s + (−0.104 − 0.994i)19-s + (−0.393 − 0.919i)20-s + (−0.753 − 0.657i)22-s + (0.599 + 0.800i)23-s + (0.251 + 0.967i)25-s + (−0.0149 − 0.999i)26-s + ⋯ |
L(s) = 1 | + (−0.251 + 0.967i)2-s + (−0.873 − 0.486i)4-s + (0.791 + 0.611i)5-s + (0.691 − 0.722i)8-s + (−0.791 + 0.611i)10-s + (−0.447 + 0.894i)11-s + (−0.963 + 0.266i)13-s + (0.525 + 0.850i)16-s + (−0.0149 + 0.999i)17-s + (−0.104 − 0.994i)19-s + (−0.393 − 0.919i)20-s + (−0.753 − 0.657i)22-s + (0.599 + 0.800i)23-s + (0.251 + 0.967i)25-s + (−0.0149 − 0.999i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3360140974 + 0.5309905086i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3360140974 + 0.5309905086i\) |
\(L(1)\) |
\(\approx\) |
\(0.6234060163 + 0.5366361792i\) |
\(L(1)\) |
\(\approx\) |
\(0.6234060163 + 0.5366361792i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.251 + 0.967i)T \) |
| 5 | \( 1 + (0.791 + 0.611i)T \) |
| 11 | \( 1 + (-0.447 + 0.894i)T \) |
| 13 | \( 1 + (-0.963 + 0.266i)T \) |
| 17 | \( 1 + (-0.0149 + 0.999i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.599 + 0.800i)T \) |
| 29 | \( 1 + (-0.995 + 0.0896i)T \) |
| 31 | \( 1 + (0.978 - 0.207i)T \) |
| 37 | \( 1 + (0.420 - 0.907i)T \) |
| 43 | \( 1 + (0.983 + 0.178i)T \) |
| 47 | \( 1 + (-0.251 + 0.967i)T \) |
| 53 | \( 1 + (-0.873 - 0.486i)T \) |
| 59 | \( 1 + (-0.337 - 0.941i)T \) |
| 61 | \( 1 + (-0.992 - 0.119i)T \) |
| 67 | \( 1 + (-0.669 + 0.743i)T \) |
| 71 | \( 1 + (-0.995 - 0.0896i)T \) |
| 73 | \( 1 + (-0.365 + 0.930i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.712 + 0.701i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−17.25740742050800547682177633053, −16.7273877552097351597650110934, −16.35840439513807684863339373187, −15.23294409905588718304040455250, −14.35341418386092397713230525943, −13.79636548084135930447602207333, −13.25847683423800467784520877619, −12.57798000687638379171930064401, −12.04815361100185204637945312073, −11.33910133171668425646816175397, −10.41642585811294801948397562355, −10.13826843708217651649619539502, −9.26842466399553369272520139218, −8.857053052599511833410954902949, −8.02463496235841611246428644092, −7.4588508524311489353222410663, −6.26700737904187285189540413855, −5.536991761510807425126675783176, −4.87072898648689514663522871098, −4.32408394057256682820280597042, −3.10569711905395289209750309969, −2.7328819839509710842994584236, −1.86379348765503439110353867993, −1.03247542947671608495093310989, −0.18399374213418070898753215737,
1.29236601918232795947309509100, 2.10427607325417283793962558096, 2.870835334495779574093031388481, 3.98998160956727593711836204716, 4.77419385882645228968274109669, 5.3348486645262505137389616070, 6.14816941116135119027618771730, 6.69162273767267378514145387471, 7.50999402900722396994026090228, 7.74185745220616022094184980283, 9.03755010100022400212140656705, 9.37882678286378028207169916842, 10.02741882204571349124596423754, 10.668621376132272591036431430701, 11.36367708137514835017102859457, 12.6216085027093576336166757460, 12.976188801717857416997207617264, 13.706041753928700232862369920671, 14.44511368236755010109090607549, 14.953408216284481896736608013, 15.36855396082641593949933290494, 16.18149565549221424421503502150, 17.113018385069677089127132085231, 17.51450531570823300736896655219, 17.74748761165709582218335012607