L(s) = 1 | + (0.951 − 0.309i)2-s + (1.04 + 1.37i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (1.42 + 0.987i)6-s + (0.426 + 4.05i)7-s + (0.587 − 0.809i)8-s + (−0.801 + 2.89i)9-s + (0.669 − 0.743i)10-s + (3.30 + 1.47i)11-s + (1.65 + 0.498i)12-s + (−0.315 − 1.48i)13-s + (1.66 + 3.72i)14-s + (1.59 + 0.669i)15-s + (0.309 − 0.951i)16-s + (−0.831 + 0.370i)17-s + ⋯ |
L(s) = 1 | + (0.672 − 0.218i)2-s + (0.605 + 0.795i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (0.581 + 0.402i)6-s + (0.161 + 1.53i)7-s + (0.207 − 0.286i)8-s + (−0.267 + 0.963i)9-s + (0.211 − 0.235i)10-s + (0.996 + 0.443i)11-s + (0.478 + 0.144i)12-s + (−0.0875 − 0.411i)13-s + (0.443 + 0.996i)14-s + (0.412 + 0.172i)15-s + (0.0772 − 0.237i)16-s + (−0.201 + 0.0897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.74725 + 1.43747i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.74725 + 1.43747i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (-1.04 - 1.37i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-3.18 + 4.56i)T \) |
good | 7 | \( 1 + (-0.426 - 4.05i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.30 - 1.47i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.315 + 1.48i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.831 - 0.370i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (5.99 + 1.27i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (5.09 + 3.70i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.234 - 0.722i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-6.51 - 3.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.77 - 4.30i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.06 + 9.73i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-9.32 - 3.03i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.09 - 10.4i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.39 + 5.75i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 10.8iT - 61T^{2} \) |
| 67 | \( 1 + (3.17 + 5.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.69 - 0.283i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (4.41 - 9.91i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (5.32 + 11.9i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.85 + 5.39i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-5.08 + 3.69i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 7.32i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.14957261876102528393909205144, −9.312088903829125707469320623813, −8.761970240873798100022930023301, −7.906205693444784924755697327645, −6.36730166483254477770168324271, −5.78776528494934197004184342417, −4.69298550682685169449944994254, −4.07845758599737151135172320031, −2.64580317599010601910058885121, −2.07915039695425900841929100754,
1.20990429319985279533684805853, 2.40994416692799664025516384895, 3.81494802729773257656369545198, 4.21079626528280977081025410714, 5.90572421469171505376512176179, 6.58324110796964350318461713220, 7.23268050180281053101752094192, 8.033473078364333587711117615941, 8.958117178752423325182625614175, 9.958037357143145441728953028177