| L(s) = 1 | + (2.92 + 3.66i)2-s + (−13.3 − 2.01i)3-s + (2.22 − 9.73i)4-s + (69.6 + 47.4i)5-s + (−31.6 − 54.8i)6-s + (−99.9 + 173. i)7-s + (177. − 85.4i)8-s + (−57.8 − 17.8i)9-s + (29.5 + 394. i)10-s + (142. + 622. i)11-s + (−49.2 + 125. i)12-s + (−27.8 + 372. i)13-s + (−927. + 139. i)14-s + (−834. − 774. i)15-s + (545. + 262. i)16-s + (1.10e3 − 755. i)17-s + ⋯ |
| L(s) = 1 | + (0.517 + 0.648i)2-s + (−0.856 − 0.129i)3-s + (0.0694 − 0.304i)4-s + (1.24 + 0.849i)5-s + (−0.359 − 0.622i)6-s + (−0.771 + 1.33i)7-s + (0.980 − 0.472i)8-s + (−0.238 − 0.0734i)9-s + (0.0934 + 1.24i)10-s + (0.353 + 1.55i)11-s + (−0.0987 + 0.251i)12-s + (−0.0457 + 0.610i)13-s + (−1.26 + 0.190i)14-s + (−0.957 − 0.888i)15-s + (0.532 + 0.256i)16-s + (0.930 − 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.13934 + 1.37053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13934 + 1.37053i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.17e4 - 3.03e3i)T \) |
| good | 2 | \( 1 + (-2.92 - 3.66i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (13.3 + 2.01i)T + (232. + 71.6i)T^{2} \) |
| 5 | \( 1 + (-69.6 - 47.4i)T + (1.14e3 + 2.90e3i)T^{2} \) |
| 7 | \( 1 + (99.9 - 173. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-142. - 622. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (27.8 - 372. i)T + (-3.67e5 - 5.53e4i)T^{2} \) |
| 17 | \( 1 + (-1.10e3 + 755. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (1.78e3 - 550. i)T + (2.04e6 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (-1.71e3 + 1.59e3i)T + (4.80e5 - 6.41e6i)T^{2} \) |
| 29 | \( 1 + (-1.59e3 + 240. i)T + (1.95e7 - 6.04e6i)T^{2} \) |
| 31 | \( 1 + (-2.16e3 + 5.51e3i)T + (-2.09e7 - 1.94e7i)T^{2} \) |
| 37 | \( 1 + (1.88e3 + 3.26e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (1.77e3 + 2.23e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (2.40e3 - 1.05e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (2.58e3 + 3.44e4i)T + (-4.13e8 + 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-1.77e4 - 8.54e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-9.67e3 - 2.46e4i)T + (-6.19e8 + 5.74e8i)T^{2} \) |
| 67 | \( 1 + (1.40e4 - 4.34e3i)T + (1.11e9 - 7.60e8i)T^{2} \) |
| 71 | \( 1 + (-5.28e4 - 4.89e4i)T + (1.34e8 + 1.79e9i)T^{2} \) |
| 73 | \( 1 + (-3.89e3 + 5.19e4i)T + (-2.04e9 - 3.08e8i)T^{2} \) |
| 79 | \( 1 + (-2.68e4 + 4.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.04e4 - 4.58e3i)T + (3.76e9 + 1.16e9i)T^{2} \) |
| 89 | \( 1 + (-1.40e5 - 2.11e4i)T + (5.33e9 + 1.64e9i)T^{2} \) |
| 97 | \( 1 + (-8.09e3 - 3.54e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.94922308588241234827300951884, −14.49857548704430206037436333915, −12.99509375535083900236750201622, −11.89320387053524872785503063652, −10.31069364676710621801893330921, −9.440284792856468244431055684843, −6.72757569990819561394056057050, −6.23086896705368758148354811351, −5.14794706618721177000922633560, −2.22185584432454114215556786329,
0.963289490044599520997853347544, 3.37783687825318576227927779710, 5.12490018426937547865789804643, 6.37421480609493957973127228288, 8.459640253341656425829222935102, 10.23010387722265418856485521653, 10.97504849091153110498598623087, 12.40940897201324811844792471875, 13.35500848073785573467225253538, 13.92159432211287291742611967034
