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
−11.48059601693695082287258680054, −10.86450022854068465925634146344, −9.682911273098404974947397808048, −9.096739429209811744925015113418, −7.63642700615454250600018195831, −6.75650839165848230290535772432, −5.75670843772835619177813447335, −5.06096780785574179146491865054, −2.81042278818642328523984568290, −1.97608912638688449030676923539,
1.19406582918638195211694067675, 3.08164935446251571575450347939, 4.69264177783429465070703001545, 5.40016266680739167177379883499, 6.38237499223813147705628138950, 7.78970096506374635205014125970, 8.786492370745372079970364365543, 9.853340240000326350911008598417, 10.36881206496366187664975706833, 11.20943870587854103703984502365