| 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
−12.01588016848725084438253240561, −10.17197944904978907627336057551, −9.670074952162096355531153790784, −8.777732493750778616180889836630, −8.177340644880462321771919449746, −6.70011865218338255235766303123, −6.37564332820373541530306718590, −4.48014117463995207383570184989, −3.66788702739607615431228888720, −2.08273259224923029725543213299,
0.46668846937169618821260944965, 2.33683689242122266017017891763, 3.59128378631509842038036948012, 4.06258512581595339349013012295, 6.24761578397502412581964259394, 7.17368293638980492919439230700, 8.157722252840617428642777793458, 9.162129956774172164923961658874, 9.866344623178527251291612349314, 10.57237804925686363758421824600
