| L(s) = 1 | + (−0.371 − 0.259i)2-s + (3.02 + 0.264i)3-s + (−0.613 − 1.68i)4-s + (−1.05 − 0.884i)6-s + (1.89 − 0.508i)7-s + (−0.444 + 1.66i)8-s + (6.13 + 1.08i)9-s + (0.238 + 0.412i)11-s + (−1.41 − 5.26i)12-s + (0.364 + 4.16i)13-s + (−0.836 − 0.304i)14-s + (−2.15 + 1.80i)16-s + (0.463 − 0.661i)17-s + (−1.99 − 1.99i)18-s + (−2.59 − 3.49i)19-s + ⋯ |
| L(s) = 1 | + (−0.262 − 0.183i)2-s + (1.74 + 0.152i)3-s + (−0.306 − 0.843i)4-s + (−0.430 − 0.361i)6-s + (0.717 − 0.192i)7-s + (−0.157 + 0.587i)8-s + (2.04 + 0.360i)9-s + (0.0717 + 0.124i)11-s + (−0.407 − 1.52i)12-s + (0.101 + 1.15i)13-s + (−0.223 − 0.0813i)14-s + (−0.538 + 0.451i)16-s + (0.112 − 0.160i)17-s + (−0.470 − 0.470i)18-s + (−0.596 − 0.802i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05498 - 0.624157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05498 - 0.624157i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.59 + 3.49i)T \) |
| good | 2 | \( 1 + (0.371 + 0.259i)T + (0.684 + 1.87i)T^{2} \) |
| 3 | \( 1 + (-3.02 - 0.264i)T + (2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (-1.89 + 0.508i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.238 - 0.412i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.364 - 4.16i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.463 + 0.661i)T + (-5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (-1.97 + 4.23i)T + (-14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.693 + 3.93i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (8.78 + 5.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.48 - 4.48i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.84 - 5.77i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.992 - 0.462i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (7.61 - 5.33i)T + (16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-6.51 - 3.03i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.304 - 1.72i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.30 - 1.20i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.74 - 2.48i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (0.907 - 2.49i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.0344 - 0.394i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-3.51 + 2.95i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.03 + 7.60i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.225 + 0.189i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (3.38 + 2.37i)T + (33.1 + 91.1i)T^{2} \) |
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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
−10.73065812015315472926305320086, −9.731185403496601331070634486311, −9.118399581549845052873918869986, −8.515191817154818687184826375250, −7.57195174695989737596857407375, −6.46835539430203703238159836438, −4.80387218396989359191634801527, −4.12430644955425472944497623520, −2.54892505883161423200041288199, −1.60167939236620726991922710134,
1.84806139918573702234468809012, 3.23183324951022560275235866855, 3.80445035336434858249250577935, 5.27806202045204443402486240956, 7.05892868960175512061972067215, 7.73403540369932379369779013785, 8.454272394604773290472049939076, 8.879201866539778488146801608982, 9.867346132289224625302762497381, 10.93535985254684543190775568298
