L(s) = 1 | + (0.337 − 1.37i)2-s + (−1.50 − 2.59i)3-s + (−1.77 − 0.927i)4-s + (−0.5 − 0.866i)5-s + (−4.07 + 1.18i)6-s + 4.06i·7-s + (−1.87 + 2.12i)8-s + (−3.00 + 5.20i)9-s + (−1.35 + 0.394i)10-s − 3.65i·11-s + (0.249 + 5.99i)12-s + (−1.47 − 0.854i)13-s + (5.58 + 1.37i)14-s + (−1.50 + 2.59i)15-s + (2.28 + 3.28i)16-s + (−3.34 − 5.79i)17-s + ⋯ |
L(s) = 1 | + (0.238 − 0.971i)2-s + (−0.866 − 1.50i)3-s + (−0.886 − 0.463i)4-s + (−0.223 − 0.387i)5-s + (−1.66 + 0.483i)6-s + 1.53i·7-s + (−0.661 + 0.749i)8-s + (−1.00 + 1.73i)9-s + (−0.429 + 0.124i)10-s − 1.10i·11-s + (0.0720 + 1.73i)12-s + (−0.410 − 0.236i)13-s + (1.49 + 0.366i)14-s + (−0.387 + 0.671i)15-s + (0.570 + 0.821i)16-s + (−0.810 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.251881 + 0.223720i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251881 + 0.223720i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.337 + 1.37i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.270 - 4.35i)T \) |
good | 3 | \( 1 + (1.50 + 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 4.06iT - 7T^{2} \) |
| 11 | \( 1 + 3.65iT - 11T^{2} \) |
| 13 | \( 1 + (1.47 + 0.854i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.34 + 5.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2.43 + 1.40i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.738 + 0.426i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 2.39iT - 37T^{2} \) |
| 41 | \( 1 + (7.85 - 4.53i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.53 - 2.61i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (11.5 + 6.64i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.43 - 1.40i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.0742 - 0.128i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.45 + 4.24i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.62 + 2.80i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.02 - 10.4i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.443 + 0.767i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.61 + 6.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 + (4.29 + 2.47i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.44 - 3.72i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.28903800483768302420264752964, −9.877894460833960644335563385577, −8.656788325926011818116231430667, −8.100853115261311261525897879980, −6.52455201379152281236134383747, −5.67812744370592178184527809132, −4.97110048832096366567651025474, −2.93534155713263956723814383887, −1.84039757004204972814438052065, −0.22715973755761205379622359042,
3.68772889153539855897016766635, 4.35269019736447459352957209640, 5.00582122935263351443277620778, 6.45307434633557938788610599569, 7.02873677152159569780179877294, 8.278247116727905593110691181909, 9.588648250932447703004908819381, 10.15281714798059158486626300509, 10.90118688836715901829354621914, 11.90871699274495476992755198608