L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.104 + 0.994i)3-s + (−0.978 + 0.207i)4-s + (−0.5 − 0.866i)5-s + 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.809 + 0.587i)10-s + (0.309 − 0.951i)11-s + (−0.104 − 0.994i)12-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (−0.104 + 0.994i)18-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.104 + 0.994i)3-s + (−0.978 + 0.207i)4-s + (−0.5 − 0.866i)5-s + 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.809 + 0.587i)10-s + (0.309 − 0.951i)11-s + (−0.104 − 0.994i)12-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)15-s + (0.913 − 0.406i)16-s + (0.309 + 0.951i)17-s + (−0.104 + 0.994i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.03180079663 - 0.4276609149i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.03180079663 - 0.4276609149i\) |
\(L(1)\) |
\(\approx\) |
\(0.5659880706 - 0.3379155709i\) |
\(L(1)\) |
\(\approx\) |
\(0.5659880706 - 0.3379155709i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.978 - 0.207i)T \) |
| 29 | \( 1 + (-0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (-0.809 + 0.587i)T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 + (-0.978 + 0.207i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.913 + 0.406i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.978 + 0.207i)T \) |
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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
−24.84058404771390913088732383900, −23.9568627231221559992656347464, −23.28032611393759798891035354752, −22.47813019995034964092890675555, −21.84894265919945151618203137632, −20.15828073830506106820445855832, −19.20272714225074548045038645009, −18.432827089229833214152601838342, −17.99633427778995450220252695841, −17.118538545890183568063813556522, −15.87979208321032581278735597973, −15.0237134514750412985979924126, −14.4278913722776352867675402683, −13.483049324612378535969466593705, −12.29801879369959132978003206211, −11.71376413448873446587726802300, −10.351019878141676170081707768875, −9.074923227153584028104173308639, −8.19971040727226871809568232356, −7.264711961881482527556961742356, −6.69881428923520385291974511029, −5.68878504429936899618868090960, −4.60558964263528282476295706352, −3.052749708458097931994200575339, −1.773765075259592803944277525583,
0.26643109403937925140027665973, 1.68252365611185411394468080577, 3.52649132727243363971114128661, 3.97645509661355245585638839559, 4.85493155464601861117877511546, 6.00145552872613933063933161354, 8.12777991550810561269714386097, 8.433173303325498111777894536075, 9.67045753056790694062576009769, 10.44171790131958359953167147017, 11.26444305572193188454872144819, 12.01835131917815726711561253569, 13.1268984140373005670023658518, 14.06196556128636618951525140708, 14.95029347911671592659378583757, 16.36247378343706935738771669254, 16.82157592298360651057975443932, 17.597768340063454331824802184349, 19.07546330563271658721586426863, 19.7862834354692498833491970547, 20.46677103319735188995468479289, 21.24410586868584632926196682553, 21.79575376510792483761826763204, 23.071607619855972635624200813006, 23.48225415892892739212435866783