| L(s) = 1 | + (1.31 + 0.521i)2-s + (1.99 + 1.36i)3-s + (1.45 + 1.37i)4-s + (−3.52 − 0.263i)5-s + (1.91 + 2.83i)6-s + (−0.894 − 2.49i)7-s + (1.19 + 2.56i)8-s + (1.03 + 2.64i)9-s + (−4.49 − 2.18i)10-s + (1.94 + 0.762i)11-s + (1.03 + 4.71i)12-s + (3.29 − 2.63i)13-s + (0.123 − 3.73i)14-s + (−6.66 − 5.31i)15-s + (0.235 + 3.99i)16-s + (−4.67 − 5.03i)17-s + ⋯ |
| L(s) = 1 | + (0.929 + 0.369i)2-s + (1.15 + 0.785i)3-s + (0.727 + 0.685i)4-s + (−1.57 − 0.117i)5-s + (0.781 + 1.15i)6-s + (−0.338 − 0.941i)7-s + (0.423 + 0.906i)8-s + (0.345 + 0.880i)9-s + (−1.41 − 0.690i)10-s + (0.586 + 0.230i)11-s + (0.299 + 1.36i)12-s + (0.914 − 0.729i)13-s + (0.0331 − 0.999i)14-s + (−1.72 − 1.37i)15-s + (0.0589 + 0.998i)16-s + (−1.13 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89994 + 1.08073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89994 + 1.08073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.31 - 0.521i)T \) |
| 7 | \( 1 + (0.894 + 2.49i)T \) |
| good | 3 | \( 1 + (-1.99 - 1.36i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (3.52 + 0.263i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 0.762i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.29 + 2.63i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (4.67 + 5.03i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.13 - 1.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.95 - 3.18i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.804 + 3.52i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.522 - 0.905i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.119 - 0.0368i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-5.49 - 11.4i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.195 - 0.406i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (9.81 - 1.47i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.10 + 0.341i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.155 + 2.07i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-1.73 + 5.61i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (2.00 - 1.15i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.45 - 0.331i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.522 + 3.46i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-7.95 - 4.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.92 - 8.68i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (1.23 - 0.485i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 3.25iT - 97T^{2} \) |
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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
−12.94423809700142168399295938801, −11.67963067263272082982012530423, −10.97774870043007400634196415115, −9.588330959991218150154823750292, −8.291090871065361058492578864605, −7.73679485864827948411724312202, −6.57204240817815326666156404001, −4.54881173436485479162934065992, −3.91382404010240853258263471049, −3.14500572035053713885529263497,
2.05899509072329416389171716453, 3.40908893145501325618741857395, 4.23149097462939467669374102864, 6.24553993337940453266145160771, 7.08626093352250745625130577600, 8.395823743007743606757106074792, 8.945961749356053296232393872618, 10.84642299028706191823214862326, 11.65030515843166889901318930493, 12.48699115673349279085764831139
