| L(s) = 1 | + (−1.39 + 0.253i)2-s + (−2.72 + 1.49i)3-s + (1.87 − 0.704i)4-s + (−1.47 − 1.67i)5-s + (3.41 − 2.77i)6-s + (2.25 − 3.09i)7-s + (−2.42 + 1.45i)8-s + (3.56 − 5.62i)9-s + (2.47 + 1.96i)10-s + (1.75 − 4.43i)11-s + (−4.04 + 4.72i)12-s + (1.09 − 1.72i)13-s + (−2.34 + 4.87i)14-s + (6.53 + 2.36i)15-s + (3.00 − 2.63i)16-s + (0.691 + 0.735i)17-s + ⋯ |
| L(s) = 1 | + (−0.983 + 0.179i)2-s + (−1.57 + 0.864i)3-s + (0.935 − 0.352i)4-s + (−0.660 − 0.751i)5-s + (1.39 − 1.13i)6-s + (0.850 − 1.17i)7-s + (−0.857 + 0.514i)8-s + (1.18 − 1.87i)9-s + (0.783 + 0.620i)10-s + (0.529 − 1.33i)11-s + (−1.16 + 1.36i)12-s + (0.303 − 0.478i)13-s + (−0.627 + 1.30i)14-s + (1.68 + 0.610i)15-s + (0.751 − 0.659i)16-s + (0.167 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.186219 - 0.370836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.186219 - 0.370836i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.253i)T \) |
| 5 | \( 1 + (1.47 + 1.67i)T \) |
| good | 3 | \( 1 + (2.72 - 1.49i)T + (1.60 - 2.53i)T^{2} \) |
| 7 | \( 1 + (-2.25 + 3.09i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.75 + 4.43i)T + (-8.01 - 7.53i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 1.72i)T + (-5.53 - 11.7i)T^{2} \) |
| 17 | \( 1 + (-0.691 - 0.735i)T + (-1.06 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.453 + 0.824i)T + (-10.1 - 16.0i)T^{2} \) |
| 23 | \( 1 + (6.82 + 5.64i)T + (4.30 + 22.5i)T^{2} \) |
| 29 | \( 1 + (0.370 - 0.0706i)T + (26.9 - 10.6i)T^{2} \) |
| 31 | \( 1 + (-3.10 + 2.91i)T + (1.94 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.592 - 9.42i)T + (-36.7 + 4.63i)T^{2} \) |
| 41 | \( 1 + (3.74 + 4.53i)T + (-7.68 + 40.2i)T^{2} \) |
| 43 | \( 1 + (2.62 + 8.07i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-8.79 - 4.13i)T + (29.9 + 36.2i)T^{2} \) |
| 53 | \( 1 + (-3.92 - 0.495i)T + (51.3 + 13.1i)T^{2} \) |
| 59 | \( 1 + (-2.02 + 7.87i)T + (-51.7 - 28.4i)T^{2} \) |
| 61 | \( 1 + (-9.80 - 8.11i)T + (11.4 + 59.9i)T^{2} \) |
| 67 | \( 1 + (0.277 - 1.45i)T + (-62.2 - 24.6i)T^{2} \) |
| 71 | \( 1 + (2.96 - 6.29i)T + (-45.2 - 54.7i)T^{2} \) |
| 73 | \( 1 + (3.21 + 12.5i)T + (-63.9 + 35.1i)T^{2} \) |
| 79 | \( 1 + (4.64 - 2.55i)T + (42.3 - 66.7i)T^{2} \) |
| 83 | \( 1 + (10.5 + 5.77i)T + (44.4 + 70.0i)T^{2} \) |
| 89 | \( 1 + (7.88 - 2.02i)T + (77.9 - 42.8i)T^{2} \) |
| 97 | \( 1 + (-12.1 + 2.32i)T + (90.1 - 35.7i)T^{2} \) |
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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
−10.05020200209533090822572440801, −8.730446642755892588407930717005, −8.221833623977926337417520457997, −7.17487699545362524438095974605, −6.18947300244193003306813665268, −5.48582889320276204078127960967, −4.45646920346996204401568263918, −3.69729136990931831860250905428, −1.10682810755521733335012478884, −0.40812503929829009723705258478,
1.49487151558882767144359162805, 2.28199075573123457034322029649, 4.12962642704879734311572332549, 5.43290118142786877926772264794, 6.22761122564447337850473266367, 7.03989427672190670414055144820, 7.58730815950866810565391074172, 8.403830048150516903108537003482, 9.626913843660957814441128792218, 10.40354650358083371060354773457
