| L(s) = 1 | + (1.29 + 1.02i)2-s + (−0.814 − 0.580i)3-s + (0.171 + 0.707i)4-s + (0.682 − 0.131i)5-s + (−0.465 − 1.58i)6-s + (2.26 + 1.36i)7-s + (0.872 − 1.91i)8-s + (0.327 + 0.945i)9-s + (1.02 + 0.526i)10-s + (−2.09 − 2.66i)11-s + (0.270 − 0.676i)12-s + (0.864 + 1.34i)13-s + (1.54 + 4.08i)14-s + (−0.632 − 0.288i)15-s + (4.37 − 2.25i)16-s + (3.99 + 3.81i)17-s + ⋯ |
| L(s) = 1 | + (0.918 + 0.721i)2-s + (−0.470 − 0.334i)3-s + (0.0858 + 0.353i)4-s + (0.305 − 0.0588i)5-s + (−0.189 − 0.647i)6-s + (0.856 + 0.516i)7-s + (0.308 − 0.675i)8-s + (0.109 + 0.315i)9-s + (0.322 + 0.166i)10-s + (−0.631 − 0.803i)11-s + (0.0781 − 0.195i)12-s + (0.239 + 0.373i)13-s + (0.412 + 1.09i)14-s + (−0.163 − 0.0745i)15-s + (1.09 − 0.564i)16-s + (0.970 + 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.21200 + 0.349221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21200 + 0.349221i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 + 0.580i)T \) |
| 7 | \( 1 + (-2.26 - 1.36i)T \) |
| 23 | \( 1 + (-3.31 + 3.46i)T \) |
| good | 2 | \( 1 + (-1.29 - 1.02i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.682 + 0.131i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (2.09 + 2.66i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.864 - 1.34i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.99 - 3.81i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.05 + 3.86i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (2.35 - 0.692i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.692 - 7.25i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (8.00 - 2.77i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (-8.73 + 7.57i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (6.54 - 2.98i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (1.10 + 0.640i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.75 + 0.464i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (3.95 - 7.66i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (2.25 + 3.16i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (0.828 + 2.06i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (-2.03 + 14.1i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (15.6 - 3.78i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-11.8 + 0.564i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (6.10 - 7.04i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (11.2 + 1.07i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (3.89 + 4.49i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.10571284749390816324059152220, −10.37013120042693304773874364978, −9.089147558168081617201367352593, −8.053211538627721748515784470559, −7.15220495692069933049517935548, −6.08564957569615423892981123485, −5.43068221380571623615755087991, −4.77430500128353028774035698978, −3.27123789072306345351210028753, −1.42767310337510951344058938334,
1.61953315662085363168073550310, 3.09081091809230621640114795811, 4.17501252178837286854374836530, 5.13702818735938417452050892083, 5.68453701382076941156932060765, 7.44625898273574688712860899621, 7.962556570447102091638270778784, 9.619636643352851275403955357557, 10.23934254220578162369002780648, 11.21710554953141501226620575982
