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
−11.21710554953141501226620575982, −10.23934254220578162369002780648, −9.619636643352851275403955357557, −7.962556570447102091638270778784, −7.44625898273574688712860899621, −5.68453701382076941156932060765, −5.13702818735938417452050892083, −4.17501252178837286854374836530, −3.09081091809230621640114795811, −1.61953315662085363168073550310,
1.42767310337510951344058938334, 3.27123789072306345351210028753, 4.77430500128353028774035698978, 5.43068221380571623615755087991, 6.08564957569615423892981123485, 7.15220495692069933049517935548, 8.053211538627721748515784470559, 9.089147558168081617201367352593, 10.37013120042693304773874364978, 11.10571284749390816324059152220