| L(s) = 1 | + (−0.874 − 1.11i)2-s + (−1.08 + 1.34i)3-s + (−0.472 + 1.94i)4-s + (3.36 − 1.56i)5-s + (2.44 + 0.0341i)6-s + (−4.13 − 0.728i)7-s + (2.57 − 1.17i)8-s + (−0.626 − 2.93i)9-s + (−4.68 − 2.36i)10-s + (−0.341 + 0.731i)11-s + (−2.10 − 2.75i)12-s + (1.91 + 0.167i)13-s + (2.80 + 5.22i)14-s + (−1.55 + 6.23i)15-s + (−3.55 − 1.83i)16-s + (−1.85 − 3.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.618 − 0.786i)2-s + (−0.628 + 0.777i)3-s + (−0.236 + 0.971i)4-s + (1.50 − 0.701i)5-s + (0.999 + 0.0139i)6-s + (−1.56 − 0.275i)7-s + (0.909 − 0.414i)8-s + (−0.208 − 0.977i)9-s + (−1.48 − 0.748i)10-s + (−0.102 + 0.220i)11-s + (−0.607 − 0.794i)12-s + (0.531 + 0.0464i)13-s + (0.748 + 1.39i)14-s + (−0.400 + 1.61i)15-s + (−0.888 − 0.458i)16-s + (−0.450 − 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0861 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.599825 - 0.550196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.599825 - 0.550196i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.874 + 1.11i)T \) |
| 3 | \( 1 + (1.08 - 1.34i)T \) |
| good | 5 | \( 1 + (-3.36 + 1.56i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (4.13 + 0.728i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.341 - 0.731i)T + (-7.07 - 8.42i)T^{2} \) |
| 13 | \( 1 + (-1.91 - 0.167i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.85 + 3.21i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.13 + 1.64i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.09 + 0.368i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.288 + 3.29i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (1.47 + 8.39i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (5.45 + 1.46i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.16 - 6.15i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.47 + 9.59i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.730 + 4.14i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.90 - 2.90i)T - 53iT^{2} \) |
| 59 | \( 1 + (-11.3 + 5.28i)T + (37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (1.25 - 0.877i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (14.4 + 1.26i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (1.26 - 0.732i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.77 - 3.90i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.51 - 2.10i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.313 - 3.58i)T + (-81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.557 - 0.321i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.6 - 4.97i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.68405802052670205517519423684, −9.795487356591154040939690587587, −9.512587632082916143941393079568, −8.922755878393760247828296859787, −7.12884140160039360348299767178, −6.10554795529845451476446322510, −5.12535171089408197926066533249, −3.83766679644594213935016286189, −2.61466121904692807612136831500, −0.73260101433298741246808390274,
1.47858311871662912855643452664, 2.96868696840304372296626277731, 5.39850075554441845263384613280, 5.98723280297551024926670064837, 6.59842056544024416929719945146, 7.29171908598132983976428616804, 8.747414450216763243518005072857, 9.546311665407860128636332570040, 10.37434339981399999021515134845, 10.94314543173871098119048946687
