L(s) = 1 | + (0.323 − 0.559i)2-s + (−1.03 − 1.79i)3-s + (0.790 + 1.37i)4-s + (0.5 − 0.866i)5-s − 1.34·6-s + (−0.211 − 2.63i)7-s + 2.31·8-s + (−0.655 + 1.13i)9-s + (−0.323 − 0.559i)10-s + (0.5 + 0.866i)11-s + (1.64 − 2.84i)12-s − 0.398·13-s + (−1.54 − 0.734i)14-s − 2.07·15-s + (−0.833 + 1.44i)16-s + (−2.20 − 3.82i)17-s + ⋯ |
L(s) = 1 | + (0.228 − 0.395i)2-s + (−0.599 − 1.03i)3-s + (0.395 + 0.685i)4-s + (0.223 − 0.387i)5-s − 0.548·6-s + (−0.0798 − 0.996i)7-s + 0.818·8-s + (−0.218 + 0.378i)9-s + (−0.102 − 0.177i)10-s + (0.150 + 0.261i)11-s + (0.474 − 0.821i)12-s − 0.110·13-s + (−0.412 − 0.196i)14-s − 0.536·15-s + (−0.208 + 0.360i)16-s + (−0.535 − 0.927i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.792997 - 1.17100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.792997 - 1.17100i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.211 + 2.63i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.323 + 0.559i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.03 + 1.79i)T + (-1.5 + 2.59i)T^{2} \) |
| 13 | \( 1 + 0.398T + 13T^{2} \) |
| 17 | \( 1 + (2.20 + 3.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.42 + 2.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.24 + 3.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.69T + 29T^{2} \) |
| 31 | \( 1 + (-1.17 - 2.04i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.24 - 5.62i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + (-3.78 + 6.55i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.46 - 5.99i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.32 - 5.75i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.914 + 1.58i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.61 - 4.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + (-8.15 - 14.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.03 + 3.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + (0.183 - 0.317i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−11.32078712626428744179810656022, −10.46258491244018761920459346105, −9.234073793853424271492972354585, −7.978417996203134659499762581590, −7.02585339065082168661896277905, −6.70535260805226716222884530107, −5.09043586203941912488096918948, −3.96831079163402529571304988196, −2.45555861638223335311780875619, −0.995647014130005702310698930167,
2.07295160807309207473014581961, 3.78758361257509311763870615329, 5.11036249455886125535252083138, 5.72401907687807771099289473875, 6.48667738603542736134461127000, 7.78023200796668739712316715567, 9.205995203626760925644585622818, 9.848368685262364120286633996354, 10.85871694625302366044545595971, 11.20254860459599246178402114286