| L(s) = 1 | + (−1.85 + 1.19i)2-s + (−0.989 − 0.142i)3-s + (1.19 − 2.61i)4-s + (1.10 − 0.324i)5-s + (2.00 − 0.916i)6-s + (−2.23 − 1.41i)7-s + (0.274 + 1.90i)8-s + (0.959 + 0.281i)9-s + (−1.66 + 1.92i)10-s + (−0.0983 + 0.152i)11-s + (−1.55 + 2.41i)12-s + (−3.11 − 2.70i)13-s + (5.83 − 0.0480i)14-s + (−1.14 + 0.164i)15-s + (0.978 + 1.12i)16-s + (2.24 + 4.91i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 + 0.843i)2-s + (−0.571 − 0.0821i)3-s + (0.596 − 1.30i)4-s + (0.494 − 0.145i)5-s + (0.819 − 0.374i)6-s + (−0.845 − 0.533i)7-s + (0.0969 + 0.674i)8-s + (0.319 + 0.0939i)9-s + (−0.527 + 0.608i)10-s + (−0.0296 + 0.0461i)11-s + (−0.448 + 0.697i)12-s + (−0.864 − 0.749i)13-s + (1.56 − 0.0128i)14-s + (−0.294 + 0.0423i)15-s + (0.244 + 0.282i)16-s + (0.544 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0197457 + 0.208079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0197457 + 0.208079i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.989 + 0.142i)T \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| 23 | \( 1 + (-0.108 - 4.79i)T \) |
| good | 2 | \( 1 + (1.85 - 1.19i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (-1.10 + 0.324i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (0.0983 - 0.152i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.11 + 2.70i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.24 - 4.91i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.214 + 0.469i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.47 - 7.60i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (3.48 - 0.500i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.139 - 0.473i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.65 + 9.05i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (9.24 + 1.32i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 3.91iT - 47T^{2} \) |
| 53 | \( 1 + (5.66 - 4.91i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (3.29 + 2.85i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.757 - 5.26i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (1.30 + 2.03i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (7.91 - 5.08i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.88 - 2.68i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.99 - 2.59i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (8.62 + 2.53i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.58 - 18.0i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (6.91 - 2.02i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.89485881863339430820631828571, −10.15579401149924444880856021932, −9.722786387893949877569324402465, −8.730447058745696673279972275751, −7.63407552091139361559208478413, −7.02754843810616188119469563362, −6.06354794405802672109240669735, −5.30199940447159734544775168914, −3.52060457055594203073996845820, −1.42014420505749583878975155013,
0.21334890079575942156576420653, 2.05402915280856236756701453646, 3.04914741969912868240961060806, 4.81476070948183934092942069239, 6.05866103428851907669086622306, 6.98647203400483448320464141001, 8.108722175218979813968329977349, 9.200211469677056654052916759252, 9.870632124332468744239218571866, 10.15220450073513948744040479539
