| L(s) = 1 | + (−0.0717 + 1.41i)2-s + (−1.31 + 0.129i)3-s + (−1.98 − 0.202i)4-s + (−1.48 + 2.78i)5-s + (−0.0886 − 1.87i)6-s + (0.373 − 1.87i)7-s + (0.429 − 2.79i)8-s + (−1.22 + 0.243i)9-s + (−3.82 − 2.30i)10-s + (−2.23 + 1.83i)11-s + (2.64 + 0.00899i)12-s + (−2.26 + 1.21i)13-s + (2.62 + 0.662i)14-s + (1.59 − 3.86i)15-s + (3.91 + 0.806i)16-s + (2.75 + 6.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.0507 + 0.998i)2-s + (−0.760 + 0.0749i)3-s + (−0.994 − 0.101i)4-s + (−0.665 + 1.24i)5-s + (−0.0362 − 0.763i)6-s + (0.141 − 0.710i)7-s + (0.151 − 0.988i)8-s + (−0.407 + 0.0810i)9-s + (−1.20 − 0.727i)10-s + (−0.673 + 0.552i)11-s + (0.764 + 0.00259i)12-s + (−0.628 + 0.335i)13-s + (0.702 + 0.177i)14-s + (0.413 − 0.997i)15-s + (0.979 + 0.201i)16-s + (0.667 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0199412 - 0.461905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0199412 - 0.461905i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0717 - 1.41i)T \) |
| good | 3 | \( 1 + (1.31 - 0.129i)T + (2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (1.48 - 2.78i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.373 + 1.87i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (2.23 - 1.83i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (2.26 - 1.21i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.75 - 6.64i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-4.33 - 1.31i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-0.470 - 0.314i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (0.777 - 0.947i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (3.28 + 3.28i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.08 - 6.85i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-4.65 + 6.96i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (8.37 + 0.824i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (11.2 - 4.67i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (2.90 + 3.54i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-8.29 - 4.43i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.811 + 8.23i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (0.215 + 2.19i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-6.18 - 1.22i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-2.35 - 11.8i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (2.74 + 1.13i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.28 + 4.23i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 8.44i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-6.54 - 6.54i)T + 97iT^{2} \) |
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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
−14.32457474949830252480263778983, −12.95907964535700657257577739606, −11.73343769711173988966374788376, −10.65387568730581470624189832976, −9.930150138926385217985657854493, −8.060226375475790552903911321582, −7.34517409175370694173240413464, −6.31937655731691024337732980774, −5.07560159815284717773378902561, −3.61515596763959349715060226890,
0.54774668922782752108278553158, 3.01683119381176204933572057881, 5.02042448383212418728705194454, 5.36030841117322579456432740497, 7.76399101244531153734752801244, 8.764985745810461328147468518644, 9.697570532543029423220210264062, 11.19766358345169616167616918683, 11.82259628832744169848846541781, 12.41646129155511882977057194811
