| L(s) = 1 | + (−0.987 − 5.57i)2-s + (−10.6 + 18.3i)3-s + (−30.0 + 10.9i)4-s + (88.2 − 50.9i)5-s + (112. + 40.9i)6-s + (128. − 16.6i)7-s + (90.9 + 156. i)8-s + (−104. − 180. i)9-s + (−370. − 441. i)10-s + (265. + 153. i)11-s + (116. − 669. i)12-s + 77.2i·13-s + (−219. − 699. i)14-s + 2.16e3i·15-s + (782. − 660. i)16-s + (870. + 502. i)17-s + ⋯ |
| L(s) = 1 | + (−0.174 − 0.984i)2-s + (−0.681 + 1.18i)3-s + (−0.939 + 0.343i)4-s + (1.57 − 0.911i)5-s + (1.28 + 0.464i)6-s + (0.991 − 0.128i)7-s + (0.502 + 0.864i)8-s + (−0.428 − 0.741i)9-s + (−1.17 − 1.39i)10-s + (0.660 + 0.381i)11-s + (0.234 − 1.34i)12-s + 0.126i·13-s + (−0.299 − 0.954i)14-s + 2.48i·15-s + (0.763 − 0.645i)16-s + (0.730 + 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.38287 - 0.309734i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38287 - 0.309734i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 + 5.57i)T \) |
| 7 | \( 1 + (-128. + 16.6i)T \) |
| good | 3 | \( 1 + (10.6 - 18.3i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-88.2 + 50.9i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-265. - 153. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 77.2iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-870. - 502. i)T + (7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (582. + 1.00e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-61.7 + 35.6i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 1.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (2.51e3 - 4.35e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-4.01e3 - 6.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 2.60e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 5.09e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.37e4 + 2.38e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (7.48e3 - 1.29e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.52e4 - 2.63e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.79e4 - 1.03e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-8.01e3 - 4.62e3i)T + (6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 5.03e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (3.96e4 + 2.28e4i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.91e4 + 2.25e4i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 5.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (8.03e4 - 4.63e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 6.05e4iT - 8.58e9T^{2} \) |
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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
−16.77573457456339977192424455663, −14.67446881218939433767484083923, −13.45296411686789656560433396396, −12.05279039938734711423538299551, −10.71403042413144366253006893065, −9.800082209269687706564191029072, −8.802536130859522524781643048869, −5.44282217416475570974623754224, −4.48303404805761667675564092607, −1.56075111764554779836734877909,
1.50761974781447793298615069954, 5.61224598880855367850796210504, 6.40911163288277521981891354196, 7.69689747585136520568207269550, 9.554344101407977028045640946814, 11.11318575235221562511646188382, 12.86408401131516259883156195870, 14.05506704332322255717305551769, 14.64718716153622525074474682462, 16.81561087188622923918363127776
