| L(s) = 1 | + (1.37 + 0.349i)2-s + (−1.33 + 0.333i)3-s + (1.75 + 0.959i)4-s + (−0.803 − 0.379i)5-s + (−1.94 − 0.00897i)6-s + (1.09 + 3.60i)7-s + (2.06 + 1.92i)8-s + (−0.980 + 0.523i)9-s + (−0.967 − 0.801i)10-s + (4.64 − 0.689i)11-s + (−2.65 − 0.692i)12-s + (−0.0306 + 0.0856i)13-s + (0.236 + 5.31i)14-s + (1.19 + 0.238i)15-s + (2.16 + 3.36i)16-s + (−1.84 + 0.367i)17-s + ⋯ |
| L(s) = 1 | + (0.968 + 0.247i)2-s + (−0.769 + 0.192i)3-s + (0.877 + 0.479i)4-s + (−0.359 − 0.169i)5-s + (−0.793 − 0.00366i)6-s + (0.412 + 1.36i)7-s + (0.731 + 0.681i)8-s + (−0.326 + 0.174i)9-s + (−0.305 − 0.253i)10-s + (1.40 − 0.207i)11-s + (−0.767 − 0.199i)12-s + (−0.00850 + 0.0237i)13-s + (0.0632 + 1.42i)14-s + (0.309 + 0.0614i)15-s + (0.540 + 0.841i)16-s + (−0.448 + 0.0891i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44800 + 0.954613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44800 + 0.954613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 - 0.349i)T \) |
| good | 3 | \( 1 + (1.33 - 0.333i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (0.803 + 0.379i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-1.09 - 3.60i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-4.64 + 0.689i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (0.0306 - 0.0856i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (1.84 - 0.367i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (2.45 + 2.22i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-6.21 + 0.612i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (2.37 + 3.20i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (6.01 + 2.49i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.000969 + 0.0197i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (2.48 - 3.02i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-1.65 + 6.59i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-4.60 - 3.07i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-7.40 + 9.98i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-2.49 - 6.97i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-4.64 + 7.75i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (8.91 + 5.34i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (1.38 - 2.58i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-1.71 + 5.65i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-8.27 - 12.3i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.409 - 8.34i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (16.6 + 1.63i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-0.260 + 0.628i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.98924233277042835695085905384, −11.55840085561476161317372613337, −10.87746238525408934906547130133, −9.091554447096420408602025764986, −8.307037979987519133479219418688, −6.81796025780076198219714848239, −5.91605116823233765491804326410, −5.10332858510947079772135441053, −4.00533657355826316918388612859, −2.33740422489871655348133667819,
1.31107099175535242383815156095, 3.53710366905520409096339025075, 4.39435961045705946466879842999, 5.64209935359852623371693495524, 6.81714842470411312240172018192, 7.30170518810376371954561198894, 9.058730581316050784592949597225, 10.51674553545155289388913827554, 11.13210635278696298167562411011, 11.75278348843173681274137551066
