L(s) = 1 | + (−0.0760 + 1.73i)3-s + (−1.38 + 3.79i)5-s + (−1.08 − 0.192i)7-s + (−2.98 − 0.263i)9-s + (5.69 − 2.07i)11-s + (−2.30 + 1.93i)13-s + (−6.46 − 2.68i)15-s + (−2.73 + 1.57i)17-s + (−3.28 − 1.89i)19-s + (0.415 − 1.87i)21-s + (0.347 + 1.96i)23-s + (−8.69 − 7.29i)25-s + (0.683 − 5.15i)27-s + (2.14 − 2.56i)29-s + (−3.05 + 0.538i)31-s + ⋯ |
L(s) = 1 | + (−0.0439 + 0.999i)3-s + (−0.618 + 1.69i)5-s + (−0.411 − 0.0725i)7-s + (−0.996 − 0.0877i)9-s + (1.71 − 0.624i)11-s + (−0.639 + 0.536i)13-s + (−1.67 − 0.692i)15-s + (−0.663 + 0.382i)17-s + (−0.753 − 0.435i)19-s + (0.0906 − 0.408i)21-s + (0.0723 + 0.410i)23-s + (−1.73 − 1.45i)25-s + (0.131 − 0.991i)27-s + (0.399 − 0.475i)29-s + (−0.548 + 0.0967i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0849993 + 0.892015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0849993 + 0.892015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0760 - 1.73i)T \) |
good | 5 | \( 1 + (1.38 - 3.79i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.08 + 0.192i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-5.69 + 2.07i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.30 - 1.93i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.28 + 1.89i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.347 - 1.96i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.14 + 2.56i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.05 - 0.538i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.96 - 8.60i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.58 - 6.66i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.77 - 4.86i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.132 + 0.753i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 3.19iT - 53T^{2} \) |
| 59 | \( 1 + (7.48 + 2.72i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.13 - 6.45i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.17 - 3.78i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.30 - 7.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.12 + 7.14i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.29 + 5.11i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.12 - 7.65i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.9 - 6.31i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.13 - 2.23i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.33049801586429935351886177608, −10.84247889118151782442371863704, −9.803286509144479530183205346802, −9.121136901349513025104289611420, −7.926754598415336472305023745895, −6.51111100491751797813479354809, −6.39300757966573605894076332023, −4.42598581282222759498105050564, −3.68688809096466986169703195595, −2.70994824455377224062808324787,
0.56946666388796811530359699386, 1.95699834398419557649999991584, 3.87377018490582468122826368527, 4.87430516372036808108085029339, 6.06229439070826279716356699814, 7.08397154256056998729883204121, 7.978387511193795883419254505010, 8.976817317862257782861379654408, 9.345697215830444228573140362679, 11.01804095276527226999050173134