L(s) = 1 | + (1 + 1.73i)2-s + (0.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−0.5 − 0.866i)5-s + 1.99·6-s + (1 + 1.73i)9-s + (0.999 − 1.73i)10-s + (−0.5 + 0.866i)11-s + (1 + 1.73i)12-s + 4·13-s − 0.999·15-s + (1.99 + 3.46i)16-s + (1 − 1.73i)17-s + (−2 + 3.46i)18-s + 1.99·20-s + ⋯ |
L(s) = 1 | + (0.707 + 1.22i)2-s + (0.288 − 0.499i)3-s + (−0.499 + 0.866i)4-s + (−0.223 − 0.387i)5-s + 0.816·6-s + (0.333 + 0.577i)9-s + (0.316 − 0.547i)10-s + (−0.150 + 0.261i)11-s + (0.288 + 0.499i)12-s + 1.10·13-s − 0.258·15-s + (0.499 + 0.866i)16-s + (0.242 − 0.420i)17-s + (−0.471 + 0.816i)18-s + 0.447·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99710 + 1.32843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99710 + 1.32843i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−10.99960232406387512276992073601, −10.11208183022591151771880337247, −8.648243581903823590806133116743, −8.160739886601978036544926623006, −7.18859000729023107478828220508, −6.58650405740226903603653941299, −5.38491769674850279670869745872, −4.68830235543696078580726996581, −3.47857556716492841054601252798, −1.64381935873697940528143599926,
1.43200026555227946777253848653, 3.03734137867374490769487646022, 3.63787617343264988492771006092, 4.50798153946797896833286686474, 5.76401693069603270053043269476, 6.94202341810636088523073471072, 8.160990163992078280327318046859, 9.187806742001175353798185728071, 10.10161325232878934977793905390, 10.79442945066406867786427534713