L(s) = 1 | + (−0.5 + 0.866i)3-s + (3 − 1.73i)5-s + (0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−3 − 1.73i)11-s − 1.73i·13-s + 3.46i·15-s + (2.5 + 4.33i)19-s + (2 + 1.73i)21-s + (6 − 3.46i)23-s + (3.5 − 6.06i)25-s + 0.999·27-s + (−2.5 + 4.33i)31-s + (3 − 1.73i)33-s + (−3 − 8.66i)35-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (1.34 − 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.904 − 0.522i)11-s − 0.480i·13-s + 0.894i·15-s + (0.573 + 0.993i)19-s + (0.436 + 0.377i)21-s + (1.25 − 0.722i)23-s + (0.700 − 1.21i)25-s + 0.192·27-s + (−0.449 + 0.777i)31-s + (0.522 − 0.301i)33-s + (−0.507 − 1.46i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38654 - 0.419032i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38654 - 0.419032i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6 + 3.46i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 8.66iT - 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 18T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−11.20943870587854103703984502365, −10.36881206496366187664975706833, −9.853340240000326350911008598417, −8.786492370745372079970364365543, −7.78970096506374635205014125970, −6.38237499223813147705628138950, −5.40016266680739167177379883499, −4.69264177783429465070703001545, −3.08164935446251571575450347939, −1.19406582918638195211694067675,
1.97608912638688449030676923539, 2.81042278818642328523984568290, 5.06096780785574179146491865054, 5.75670843772835619177813447335, 6.75650839165848230290535772432, 7.63642700615454250600018195831, 9.096739429209811744925015113418, 9.682911273098404974947397808048, 10.86450022854068465925634146344, 11.48059601693695082287258680054