L(s) = 1 | + 0.369i·2-s + (0.867 − 1.49i)3-s + 1.86·4-s + 1.03·5-s + (0.554 + 0.320i)6-s − i·7-s + 1.42i·8-s + (−1.49 − 2.60i)9-s + 0.381i·10-s − 0.430·11-s + (1.61 − 2.79i)12-s + 4.27·13-s + 0.369·14-s + (0.895 − 1.54i)15-s + 3.19·16-s − 7.25·17-s + ⋯ |
L(s) = 1 | + 0.261i·2-s + (0.500 − 0.865i)3-s + 0.931·4-s + 0.461·5-s + (0.226 + 0.130i)6-s − 0.377i·7-s + 0.504i·8-s + (−0.498 − 0.866i)9-s + 0.120i·10-s − 0.129·11-s + (0.466 − 0.806i)12-s + 1.18·13-s + 0.0987·14-s + (0.231 − 0.399i)15-s + 0.799·16-s − 1.75·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.06060 - 0.627889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06060 - 0.627889i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.867 + 1.49i)T \) |
| 7 | \( 1 + iT \) |
| 23 | \( 1 + (-4.78 - 0.321i)T \) |
good | 2 | \( 1 - 0.369iT - 2T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 11 | \( 1 + 0.430T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 - 3.53iT - 19T^{2} \) |
| 29 | \( 1 + 4.52iT - 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 8.22iT - 37T^{2} \) |
| 41 | \( 1 - 4.85iT - 41T^{2} \) |
| 43 | \( 1 + 9.90iT - 43T^{2} \) |
| 47 | \( 1 - 0.312iT - 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 8.46iT - 59T^{2} \) |
| 61 | \( 1 + 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 5.50iT - 67T^{2} \) |
| 71 | \( 1 - 12.5iT - 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 - 17.2iT - 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 4.36T + 89T^{2} \) |
| 97 | \( 1 - 4.63iT - 97T^{2} \) |
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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
−11.09405635000096957609328879930, −10.04255365795241387396472636275, −8.817495588952654473904385119779, −8.132361894723861190848270332076, −7.08988442631063782944918999168, −6.46798925356612058712184118073, −5.65860991556390662678821910609, −3.84779009380880535740155126532, −2.54638098813064946619267584560, −1.50285009225744959367436864128,
1.98093549153666650178242548484, 2.94470146923867936797562001067, 4.11019290073275151206917187462, 5.40695936248420105606299954478, 6.37492099174348160581221291584, 7.41523515039944010526742521831, 8.778611665489157103806217802435, 9.143835463800743958999342247142, 10.38511837080398025396667811102, 11.00034286712378235997670693572