| L(s) = 1 | + (−0.143 + 1.40i)2-s + (0.881 + 0.471i)3-s + (−1.95 − 0.403i)4-s + (−0.269 + 0.328i)5-s + (−0.789 + 1.17i)6-s + (3.15 + 2.11i)7-s + (0.848 − 2.69i)8-s + (0.555 + 0.831i)9-s + (−0.424 − 0.426i)10-s + (3.14 + 0.953i)11-s + (−1.53 − 1.27i)12-s + (−1.62 + 1.32i)13-s + (−3.42 + 4.14i)14-s + (−0.393 + 0.162i)15-s + (3.67 + 1.58i)16-s + (−2.65 − 1.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.101 + 0.994i)2-s + (0.509 + 0.272i)3-s + (−0.979 − 0.201i)4-s + (−0.120 + 0.147i)5-s + (−0.322 + 0.478i)6-s + (1.19 + 0.797i)7-s + (0.299 − 0.953i)8-s + (0.185 + 0.277i)9-s + (−0.134 − 0.134i)10-s + (0.947 + 0.287i)11-s + (−0.443 − 0.369i)12-s + (−0.449 + 0.368i)13-s + (−0.914 + 1.10i)14-s + (−0.101 + 0.0420i)15-s + (0.918 + 0.395i)16-s + (−0.644 − 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.750680 + 1.28962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.750680 + 1.28962i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.143 - 1.40i)T \) |
| 3 | \( 1 + (-0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (0.269 - 0.328i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-3.15 - 2.11i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-3.14 - 0.953i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (1.62 - 1.32i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (2.65 + 1.10i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.0395 - 0.401i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (9.04 + 1.79i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 4.63i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.112 + 0.112i)T - 31iT^{2} \) |
| 37 | \( 1 + (-8.95 + 0.881i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-2.48 + 12.5i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.40 + 1.28i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.88 + 6.97i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (1.22 - 4.03i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (2.95 + 2.42i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (2.79 - 5.23i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-6.18 + 11.5i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (4.93 - 7.38i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-8.60 + 5.75i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (3.45 + 8.33i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-3.02 - 0.297i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.02 + 0.999i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-7.81 + 7.81i)T - 97iT^{2} \) |
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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
−11.71831941566731126560040195296, −10.56304552445921057930352944180, −9.364129823233212030930143528484, −8.856419386932301398703296644213, −7.940068487243220953139829111050, −7.09274981719465721754728118935, −5.90506536646160294333337108177, −4.80919435951173519766142282979, −3.96068407121167068335755174391, −1.99941569396458402414880929970,
1.12850898429451474785151941234, 2.41573467579296256735487234417, 4.00665809278758155259572169987, 4.55926912522050051281282820199, 6.22606269091666532265508246883, 7.898157357720757303035012615868, 8.100194124100700949968668453672, 9.344305803535407585128392509018, 10.14902414081096516764847225299, 11.20025019212848374212877220945
