L(s) = 1 | + (0.167 + 0.0969i)2-s + (−0.866 + 0.5i)3-s + (−0.981 − 1.69i)4-s + (2.19 − 0.445i)5-s − 0.193·6-s − 0.768i·8-s + (0.499 − 0.866i)9-s + (0.411 + 0.137i)10-s + (−1 − 1.73i)11-s + (1.69 + 0.981i)12-s + 1.35i·13-s + (−1.67 + 1.48i)15-s + (−1.88 + 3.26i)16-s + (2.90 − 1.67i)17-s + (0.167 − 0.0969i)18-s + (2.67 − 4.63i)19-s + ⋯ |
L(s) = 1 | + (0.118 + 0.0685i)2-s + (−0.499 + 0.288i)3-s + (−0.490 − 0.849i)4-s + (0.979 − 0.199i)5-s − 0.0791·6-s − 0.271i·8-s + (0.166 − 0.288i)9-s + (0.130 + 0.0435i)10-s + (−0.301 − 0.522i)11-s + (0.490 + 0.283i)12-s + 0.374i·13-s + (−0.432 + 0.382i)15-s + (−0.471 + 0.817i)16-s + (0.703 − 0.406i)17-s + (0.0395 − 0.0228i)18-s + (0.613 − 1.06i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0736 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0736 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.907627 - 0.843099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.907627 - 0.843099i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-2.19 + 0.445i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.167 - 0.0969i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 + (-2.90 + 1.67i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.67 + 4.63i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.29 + 2.48i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + (2.28 + 3.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.671 + 0.387i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 + (-8.59 - 4.96i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.42 + 4.28i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.31 + 7.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.35 + 7.53i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.59 - 4.96i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (8.09 - 4.67i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.35 + 9.26i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + (-0.518 + 0.898i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 18.4iT - 97T^{2} \) |
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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
−10.16199609312863869803924213819, −9.370758151663862177415798125219, −8.896235275701272264161262754719, −7.42094311338360886288594896543, −6.30369754613979380791446285953, −5.58171148433386644803582311831, −5.05195389554258283609083414513, −3.86975992356494657428738607309, −2.18666793887950695988208025001, −0.67875154723938648913771673678,
1.66073312672501879422302646716, 3.00013456199197483998078757076, 4.14234935909360390271802037329, 5.42256519813039620609067859381, 5.89555197535336081912497746734, 7.30337106518983036399384325045, 7.78168601912770893613350945190, 8.941269188467122935213178251788, 9.833023799274751320300383603948, 10.45265803676870719395384626251