L(s) = 1 | + 0.305·2-s + (1.47 − 0.913i)3-s − 1.90·4-s − 2.05i·5-s + (0.450 − 0.279i)6-s + (−0.946 − 2.47i)7-s − 1.19·8-s + (1.33 − 2.68i)9-s − 0.628i·10-s − 5.03·11-s + (−2.80 + 1.74i)12-s + (1.75 + 3.14i)13-s + (−0.289 − 0.755i)14-s + (−1.87 − 3.02i)15-s + 3.44·16-s + 3.24·17-s + ⋯ |
L(s) = 1 | + 0.216·2-s + (0.849 − 0.527i)3-s − 0.953·4-s − 0.918i·5-s + (0.183 − 0.114i)6-s + (−0.357 − 0.933i)7-s − 0.422·8-s + (0.444 − 0.895i)9-s − 0.198i·10-s − 1.51·11-s + (−0.810 + 0.502i)12-s + (0.487 + 0.872i)13-s + (−0.0774 − 0.201i)14-s + (−0.484 − 0.780i)15-s + 0.861·16-s + 0.787·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.879621 - 1.01221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.879621 - 1.01221i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.47 + 0.913i)T \) |
| 7 | \( 1 + (0.946 + 2.47i)T \) |
| 13 | \( 1 + (-1.75 - 3.14i)T \) |
good | 2 | \( 1 - 0.305T + 2T^{2} \) |
| 5 | \( 1 + 2.05iT - 5T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 4.85iT - 37T^{2} \) |
| 41 | \( 1 + 3.35iT - 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 7.14iT - 47T^{2} \) |
| 53 | \( 1 + 3.08iT - 53T^{2} \) |
| 59 | \( 1 - 12.5iT - 59T^{2} \) |
| 61 | \( 1 + 3.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.77iT - 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 - 0.936iT - 83T^{2} \) |
| 89 | \( 1 - 11.5iT - 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.13509062009925627101518449512, −10.40499870334892830924423641326, −9.616501565309814896920313159896, −8.619200920132708029583866594852, −8.016300318622840106189869357148, −6.85874820061571939394612778966, −5.30394666721588839289253757427, −4.28558437056897545111469935254, −3.11255435706127213712509640865, −0.967952780034643072296995043895,
2.82279859132284951352313264140, 3.39418981789116250868195455148, 5.04475489917959714765985892428, 5.81915563837872536254526611201, 7.69269955451471061123433299052, 8.236279541298035508444727558006, 9.503921550419175244302965198081, 10.00226335968901875115315379026, 10.99954043795537888126213294336, 12.41562706833523082454072649093