| L(s) = 1 | + (−3.91 − 0.892i)2-s + (−5.06 + 16.4i)3-s + (0.0905 + 0.0435i)4-s + (−31.2 + 12.2i)5-s + (34.4 − 59.7i)6-s + (70.0 − 40.4i)7-s + (49.8 + 39.7i)8-s + (−177. − 120. i)9-s + (133. − 20.0i)10-s + (−25.6 + 12.3i)11-s + (−1.17 + 1.26i)12-s + (−36.1 − 5.45i)13-s + (−310. + 95.6i)14-s + (−43.1 − 575. i)15-s + (−160. − 201. i)16-s + (−36.6 + 93.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.977 − 0.223i)2-s + (−0.563 + 1.82i)3-s + (0.00565 + 0.00272i)4-s + (−1.24 + 0.490i)5-s + (0.958 − 1.65i)6-s + (1.42 − 0.825i)7-s + (0.779 + 0.621i)8-s + (−2.18 − 1.49i)9-s + (1.33 − 0.200i)10-s + (−0.212 + 0.102i)11-s + (−0.00815 + 0.00879i)12-s + (−0.214 − 0.0322i)13-s + (−1.58 + 0.487i)14-s + (−0.191 − 2.55i)15-s + (−0.627 − 0.786i)16-s + (−0.126 + 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0327327 - 0.0396235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0327327 - 0.0396235i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-63.1 - 1.84e3i)T \) |
| good | 2 | \( 1 + (3.91 + 0.892i)T + (14.4 + 6.94i)T^{2} \) |
| 3 | \( 1 + (5.06 - 16.4i)T + (-66.9 - 45.6i)T^{2} \) |
| 5 | \( 1 + (31.2 - 12.2i)T + (458. - 425. i)T^{2} \) |
| 7 | \( 1 + (-70.0 + 40.4i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (25.6 - 12.3i)T + (9.12e3 - 1.14e4i)T^{2} \) |
| 13 | \( 1 + (36.1 + 5.45i)T + (2.72e4 + 8.41e3i)T^{2} \) |
| 17 | \( 1 + (36.6 - 93.4i)T + (-6.12e4 - 5.68e4i)T^{2} \) |
| 19 | \( 1 + (98.1 + 143. i)T + (-4.76e4 + 1.21e5i)T^{2} \) |
| 23 | \( 1 + (-10.9 + 146. i)T + (-2.76e5 - 4.17e4i)T^{2} \) |
| 29 | \( 1 + (46.2 + 149. i)T + (-5.84e5 + 3.98e5i)T^{2} \) |
| 31 | \( 1 + (553. + 513. i)T + (6.90e4 + 9.20e5i)T^{2} \) |
| 37 | \( 1 + (1.92e3 + 1.11e3i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-191. + 838. i)T + (-2.54e6 - 1.22e6i)T^{2} \) |
| 47 | \( 1 + (1.16e3 + 560. i)T + (3.04e6 + 3.81e6i)T^{2} \) |
| 53 | \( 1 + (4.50e3 - 679. i)T + (7.53e6 - 2.32e6i)T^{2} \) |
| 59 | \( 1 + (-2.26e3 - 2.83e3i)T + (-2.69e6 + 1.18e7i)T^{2} \) |
| 61 | \( 1 + (-1.94e3 - 2.09e3i)T + (-1.03e6 + 1.38e7i)T^{2} \) |
| 67 | \( 1 + (-4.34e3 + 2.95e3i)T + (7.36e6 - 1.87e7i)T^{2} \) |
| 71 | \( 1 + (9.99e3 - 748. i)T + (2.51e7 - 3.78e6i)T^{2} \) |
| 73 | \( 1 + (-994. + 6.59e3i)T + (-2.71e7 - 8.37e6i)T^{2} \) |
| 79 | \( 1 + (1.96e3 + 3.40e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (2.42e3 + 747. i)T + (3.92e7 + 2.67e7i)T^{2} \) |
| 89 | \( 1 + (857. - 2.78e3i)T + (-5.18e7 - 3.53e7i)T^{2} \) |
| 97 | \( 1 + (-2.28e3 + 1.09e3i)T + (5.51e7 - 6.92e7i)T^{2} \) |
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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
−15.05285709221581514579683407931, −14.34735352262792038702692017788, −11.54848943409634244934535814401, −10.94572548826461933477333770272, −10.29033702564651549713567234192, −8.836929650109868629894909441428, −7.69877100911470974671292487062, −4.93561281416797277697800928986, −4.02584980029911170348385905210, −0.05227070062954896295519552183,
1.52878308660409814249905818004, 5.10744934795810159099113899137, 7.12072684714597667406227423098, 8.104356884263051057583944740145, 8.520633048043663209961121952775, 11.14380755763902316415347580888, 11.95373110385098486734976315303, 12.83271681142763704729772537620, 14.26022208109114745771460490269, 15.89405314895837526375889538937
