L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + i·9-s + (0.809 − 0.587i)10-s + (−0.987 − 0.156i)11-s + (0.156 + 0.987i)12-s + (0.891 − 0.453i)13-s + (−0.987 + 0.156i)15-s + (0.309 + 0.951i)16-s + (−0.156 + 0.987i)17-s + (0.309 − 0.951i)18-s + (0.891 + 0.453i)19-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + i·9-s + (0.809 − 0.587i)10-s + (−0.987 − 0.156i)11-s + (0.156 + 0.987i)12-s + (0.891 − 0.453i)13-s + (−0.987 + 0.156i)15-s + (0.309 + 0.951i)16-s + (−0.156 + 0.987i)17-s + (0.309 − 0.951i)18-s + (0.891 + 0.453i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3585805969 + 0.6427876045i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3585805969 + 0.6427876045i\) |
\(L(1)\) |
\(\approx\) |
\(0.6709288113 + 0.3127360233i\) |
\(L(1)\) |
\(\approx\) |
\(0.6709288113 + 0.3127360233i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (-0.987 - 0.156i)T \) |
| 13 | \( 1 + (0.891 - 0.453i)T \) |
| 17 | \( 1 + (-0.156 + 0.987i)T \) |
| 19 | \( 1 + (0.891 + 0.453i)T \) |
| 23 | \( 1 + (-0.309 + 0.951i)T \) |
| 29 | \( 1 + (0.156 + 0.987i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (-0.951 - 0.309i)T \) |
| 47 | \( 1 + (-0.453 - 0.891i)T \) |
| 53 | \( 1 + (-0.156 - 0.987i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.951 + 0.309i)T \) |
| 67 | \( 1 + (0.987 - 0.156i)T \) |
| 71 | \( 1 + (0.987 + 0.156i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.453 + 0.891i)T \) |
| 97 | \( 1 + (0.987 - 0.156i)T \) |
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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
−25.21808736462385012302125236216, −24.42358521293151452699682690794, −23.84670269318537546906603825783, −22.9843226879412440384657405056, −20.94249260479403986780600877824, −20.48157964463060130038230791489, −19.75608371320901892463417279091, −18.59706715394404048982740035823, −18.29969377825564953461841272308, −17.03442964226273229084143226006, −15.91470763935660876580606897036, −15.47925889887186779981526765576, −14.11500242947787825086181563743, −13.18137581220035833593039388851, −12.04243300834558549610115925000, −11.19314173792180313040479095278, −9.70504653921296838914599180639, −8.89153398586389677539012307015, −8.04873243309760030592589351465, −7.38793905074382068090074790414, −6.23319279391482175088149280725, −4.846323199683465090914604665914, −3.160723129338229122832753774730, −1.902628635300361341863878891653, −0.60174237183551259951742711910,
1.85618065607998625439909005966, 3.238959757501564773982301736625, 3.63702875011099051611333233418, 5.54063597708867173778807215323, 7.088302470442140295196625960564, 8.01185796051140306133179583483, 8.639801634219542034299709105870, 9.97765430794503515876336982813, 10.615115824273212339932581213, 11.32134479688964973911181660846, 12.70988282785927560001304198275, 13.91661279464724397556134938834, 15.16455023400224899912674266241, 15.72349388550011502030544496852, 16.47624895746940909381638316888, 17.975249324283897154271077273527, 18.54928332648448863360371625337, 19.60206670284370856802163776355, 20.15548845012117435977481123916, 21.21146213723083866190612705927, 21.85004200802987131529572132310, 23.052756754231319051433056637611, 24.20400496506694313585498415395, 25.5466446033145204442985250288, 25.946069859902724315777062900102