| L(s) = 1 | + (−0.987 − 1.01i)2-s + (−1.64 + 0.537i)3-s + (−0.0490 + 1.99i)4-s + (−1.64 + 1.08i)5-s + (2.16 + 1.13i)6-s + (−3.20 − 3.02i)7-s + (2.07 − 1.92i)8-s + (2.42 − 1.76i)9-s + (2.72 + 0.597i)10-s + (0.555 + 1.10i)11-s + (−0.993 − 3.31i)12-s + (−5.40 + 2.32i)13-s + (0.104 + 6.22i)14-s + (2.12 − 2.66i)15-s + (−3.99 − 0.196i)16-s + (−0.839 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.715i)2-s + (−0.950 + 0.310i)3-s + (−0.0245 + 0.999i)4-s + (−0.736 + 0.484i)5-s + (0.885 + 0.463i)6-s + (−1.21 − 1.14i)7-s + (0.732 − 0.680i)8-s + (0.807 − 0.589i)9-s + (0.860 + 0.188i)10-s + (0.167 + 0.333i)11-s + (−0.286 − 0.958i)12-s + (−1.49 + 0.646i)13-s + (0.0279 + 1.66i)14-s + (0.549 − 0.688i)15-s + (−0.998 − 0.0490i)16-s + (−0.203 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378741 - 0.0643785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378741 - 0.0643785i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 + 1.01i)T \) |
| 3 | \( 1 + (1.64 - 0.537i)T \) |
| good | 5 | \( 1 + (1.64 - 1.08i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (3.20 + 3.02i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.555 - 1.10i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (5.40 - 2.32i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (0.839 + 1.00i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.869 - 0.729i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (2.24 + 2.38i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-9.66 + 1.12i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (0.734 + 3.09i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-4.78 + 0.843i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-8.90 - 6.63i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.456 + 7.82i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-9.28 - 2.20i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-3.53 - 6.11i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.88 + 5.74i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (7.14 - 2.13i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-6.40 - 0.748i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (1.77 + 0.646i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (7.89 - 2.87i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.76 + 4.29i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (7.58 - 5.64i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (4.69 + 12.9i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.18 - 6.04i)T + (38.4 + 89.0i)T^{2} \) |
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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
−10.40524140691802247562898970825, −9.905837587653242899090144946594, −9.246241954403528378809516467123, −7.58435082106205020951591297082, −7.16472970025105818309024912630, −6.35748212366226374363922272511, −4.47778681111895959410508104794, −3.99203844736140612498686446796, −2.71852817478620942597092895845, −0.61437004229705624661564724603,
0.57394668304344988091575930379, 2.56431854066008230994398944614, 4.48636414187119390951971202664, 5.47235370020714957649921947702, 6.14078009497513131082274733149, 7.05124224083755539708627080315, 7.893529566533422335487105105199, 8.810310069417261724939478492162, 9.719308310736837080974996340091, 10.36287469101552619574436714049
