L(s) = 1 | + (−0.0761 − 1.41i)2-s + (−0.995 − 0.0980i)3-s + (−1.98 + 0.215i)4-s + (1.41 + 2.64i)5-s + (−0.0625 + 1.41i)6-s + (−0.878 − 4.41i)7-s + (0.455 + 2.79i)8-s + (0.980 + 0.195i)9-s + (3.62 − 2.19i)10-s + (−3.15 − 2.58i)11-s + (1.99 − 0.0192i)12-s + (−2.78 − 1.48i)13-s + (−6.16 + 1.57i)14-s + (−1.14 − 2.76i)15-s + (3.90 − 0.855i)16-s + (2.08 − 5.03i)17-s + ⋯ |
L(s) = 1 | + (−0.0538 − 0.998i)2-s + (−0.574 − 0.0565i)3-s + (−0.994 + 0.107i)4-s + (0.631 + 1.18i)5-s + (−0.0255 + 0.576i)6-s + (−0.332 − 1.66i)7-s + (0.161 + 0.986i)8-s + (0.326 + 0.0650i)9-s + (1.14 − 0.694i)10-s + (−0.951 − 0.780i)11-s + (0.577 − 0.00556i)12-s + (−0.772 − 0.412i)13-s + (−1.64 + 0.421i)14-s + (−0.296 − 0.715i)15-s + (0.976 − 0.213i)16-s + (0.506 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00995644 - 0.590133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00995644 - 0.590133i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0761 + 1.41i)T \) |
| 3 | \( 1 + (0.995 + 0.0980i)T \) |
good | 5 | \( 1 + (-1.41 - 2.64i)T + (-2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (0.878 + 4.41i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (3.15 + 2.58i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.78 + 1.48i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.08 + 5.03i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (7.63 - 2.31i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.59 + 1.06i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (3.94 + 4.81i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.500 - 0.500i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.25 - 4.13i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-1.39 - 2.08i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.81 + 0.572i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (1.03 + 0.427i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.146 - 0.178i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-7.57 + 4.04i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.985 + 10.0i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (0.886 - 9.00i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-3.44 + 0.686i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.10 + 5.53i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-4.65 + 1.92i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.00 + 9.90i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (1.97 + 1.31i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-9.92 + 9.92i)T - 97iT^{2} \) |
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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
−10.67854017081370829272496453028, −10.34094499016057913032467078548, −9.667475439286545932500906277462, −7.995938257081839141872999667415, −7.13066184528700418368893756391, −6.02336028887779380001250116837, −4.78923328551516562410215281570, −3.52470144387367955513381190474, −2.44616669792081842431805765837, −0.41199533893442295183676937219,
2.07749354459203634008457516817, 4.41050444688281041591937813141, 5.37055219344052404691147963367, 5.74795226529845028376940450004, 6.90938238087888435529822467897, 8.230925661501450492803190059009, 9.009819508459638556655964048897, 9.588449362431435826058629775467, 10.65163204045282425551760979640, 12.39880375877674059590486046424