| L(s) = 1 | + (−0.451 − 1.34i)2-s + (2.95 − 0.535i)3-s + (−1.59 + 1.21i)4-s + (−1.84 + 0.733i)5-s + (−2.05 − 3.71i)6-s + (−10.2 − 4.72i)7-s + (2.34 + 1.58i)8-s + (8.42 − 3.16i)9-s + (1.81 + 2.13i)10-s + (13.7 − 3.81i)11-s + (−4.05 + 4.42i)12-s + (−5.71 − 10.7i)13-s + (−1.71 + 15.8i)14-s + (−5.04 + 3.15i)15-s + (1.07 − 3.85i)16-s + (−3.27 − 7.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 − 0.670i)2-s + (0.983 − 0.178i)3-s + (−0.398 + 0.302i)4-s + (−0.368 + 0.146i)5-s + (−0.341 − 0.619i)6-s + (−1.45 − 0.674i)7-s + (0.292 + 0.198i)8-s + (0.936 − 0.351i)9-s + (0.181 + 0.213i)10-s + (1.25 − 0.347i)11-s + (−0.337 + 0.368i)12-s + (−0.439 − 0.829i)13-s + (−0.122 + 1.12i)14-s + (−0.336 + 0.210i)15-s + (0.0668 − 0.240i)16-s + (−0.192 − 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.167408 - 1.09488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.167408 - 1.09488i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 + 1.34i)T \) |
| 3 | \( 1 + (-2.95 + 0.535i)T \) |
| 59 | \( 1 + (17.2 - 56.4i)T \) |
| good | 5 | \( 1 + (1.84 - 0.733i)T + (18.1 - 17.1i)T^{2} \) |
| 7 | \( 1 + (10.2 + 4.72i)T + (31.7 + 37.3i)T^{2} \) |
| 11 | \( 1 + (-13.7 + 3.81i)T + (103. - 62.3i)T^{2} \) |
| 13 | \( 1 + (5.71 + 10.7i)T + (-94.8 + 139. i)T^{2} \) |
| 17 | \( 1 + (3.27 + 7.08i)T + (-187. + 220. i)T^{2} \) |
| 19 | \( 1 + (23.9 + 5.27i)T + (327. + 151. i)T^{2} \) |
| 23 | \( 1 + (39.7 - 6.51i)T + (501. - 168. i)T^{2} \) |
| 29 | \( 1 + (-7.02 + 20.8i)T + (-669. - 508. i)T^{2} \) |
| 31 | \( 1 + (-38.5 + 8.49i)T + (872. - 403. i)T^{2} \) |
| 37 | \( 1 + (26.7 + 39.3i)T + (-506. + 1.27e3i)T^{2} \) |
| 41 | \( 1 + (42.1 + 6.91i)T + (1.59e3 + 536. i)T^{2} \) |
| 43 | \( 1 + (-12.4 + 44.7i)T + (-1.58e3 - 953. i)T^{2} \) |
| 47 | \( 1 + (-38.5 - 15.3i)T + (1.60e3 + 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-20.5 - 17.4i)T + (454. + 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-9.31 + 3.13i)T + (2.96e3 - 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-36.1 + 53.2i)T + (-1.66e3 - 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-12.4 - 4.97i)T + (3.65e3 + 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-116. - 12.6i)T + (5.20e3 + 1.14e3i)T^{2} \) |
| 79 | \( 1 + (16.7 + 10.0i)T + (2.92e3 + 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-119. - 6.45i)T + (6.84e3 + 744. i)T^{2} \) |
| 89 | \( 1 + (26.6 - 79.1i)T + (-6.30e3 - 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-69.1 + 7.52i)T + (9.18e3 - 2.02e3i)T^{2} \) |
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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
−10.57237804925686363758421824600, −9.866344623178527251291612349314, −9.162129956774172164923961658874, −8.157722252840617428642777793458, −7.17368293638980492919439230700, −6.24761578397502412581964259394, −4.06258512581595339349013012295, −3.59128378631509842038036948012, −2.33683689242122266017017891763, −0.46668846937169618821260944965,
2.08273259224923029725543213299, 3.66788702739607615431228888720, 4.48014117463995207383570184989, 6.37564332820373541530306718590, 6.70011865218338255235766303123, 8.177340644880462321771919449746, 8.777732493750778616180889836630, 9.670074952162096355531153790784, 10.17197944904978907627336057551, 12.01588016848725084438253240561
