| L(s) = 1 | + (0.681 + 1.23i)2-s + (1.51 − 0.149i)3-s + (−1.07 + 1.68i)4-s + (1.30 − 2.44i)5-s + (1.21 + 1.77i)6-s + (0.196 − 0.986i)7-s + (−2.82 − 0.173i)8-s + (−0.674 + 0.134i)9-s + (3.91 − 0.0477i)10-s + (−4.03 + 3.31i)11-s + (−1.36 + 2.71i)12-s + (−2.42 + 1.29i)13-s + (1.35 − 0.429i)14-s + (1.61 − 3.88i)15-s + (−1.70 − 3.61i)16-s + (−1.15 − 2.78i)17-s + ⋯ |
| L(s) = 1 | + (0.482 + 0.876i)2-s + (0.873 − 0.0860i)3-s + (−0.535 + 0.844i)4-s + (0.583 − 1.09i)5-s + (0.496 + 0.723i)6-s + (0.0741 − 0.373i)7-s + (−0.998 − 0.0614i)8-s + (−0.224 + 0.0447i)9-s + (1.23 − 0.0151i)10-s + (−1.21 + 0.998i)11-s + (−0.394 + 0.784i)12-s + (−0.672 + 0.359i)13-s + (0.362 − 0.114i)14-s + (0.415 − 1.00i)15-s + (−0.427 − 0.904i)16-s + (−0.279 − 0.674i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45816 + 0.635333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45816 + 0.635333i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.681 - 1.23i)T \) |
| good | 3 | \( 1 + (-1.51 + 0.149i)T + (2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (-1.30 + 2.44i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.196 + 0.986i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (4.03 - 3.31i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (2.42 - 1.29i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (1.15 + 2.78i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.73 - 0.829i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-4.27 - 2.85i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-6.36 + 7.75i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.52 - 4.52i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.68 + 5.56i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (1.13 - 1.70i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.815 - 0.0803i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 0.516i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (7.04 + 8.57i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (5.87 + 3.14i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.787 - 7.99i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-1.20 - 12.1i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (1.46 + 0.290i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 6.89i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (4.56 + 1.88i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.22 + 7.31i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-8.41 + 5.62i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-4.62 - 4.62i)T + 97iT^{2} \) |
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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
−13.57589409436744228940126336996, −12.93586820086110524930821552774, −11.88648929694984481436027675669, −9.883253498495759840228659834575, −9.055705442254182227494892754053, −8.030280421617847007566164902034, −7.16261137222616333879246840918, −5.40078515190845962561405172594, −4.60365266584767701594360448490, −2.67057194286562910991897782974,
2.63375349267133395125855514353, 3.08025937601029616257755235115, 5.15454046859804635071485204946, 6.34778999939655875835891315153, 8.133440361679072779782627474490, 9.183190852505682683561180290221, 10.38193138685573799629368320347, 10.92067004536791789780845083310, 12.27104524377843575831560218806, 13.46497058316367438849790863072
