| L(s) = 1 | + (−0.326 − 1.37i)2-s + (−1.15 − 2.76i)3-s + (−1.78 + 0.897i)4-s + (5.90 − 4.39i)5-s + (−3.43 + 2.49i)6-s + (11.1 − 7.31i)7-s + (1.81 + 2.16i)8-s + (−6.33 + 6.39i)9-s + (−7.96 − 6.68i)10-s + (−1.19 + 10.2i)11-s + (4.54 + 3.91i)12-s + (−2.19 − 7.31i)13-s + (−13.6 − 12.9i)14-s + (−18.9 − 11.2i)15-s + (2.38 − 3.20i)16-s + (−10.5 − 1.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.163 − 0.688i)2-s + (−0.384 − 0.923i)3-s + (−0.446 + 0.224i)4-s + (1.18 − 0.878i)5-s + (−0.572 + 0.415i)6-s + (1.58 − 1.04i)7-s + (0.227 + 0.270i)8-s + (−0.704 + 0.710i)9-s + (−0.796 − 0.668i)10-s + (−0.108 + 0.932i)11-s + (0.379 + 0.326i)12-s + (−0.168 − 0.562i)13-s + (−0.978 − 0.922i)14-s + (−1.26 − 0.751i)15-s + (0.149 − 0.200i)16-s + (−0.618 − 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.527917 - 1.38927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.527917 - 1.38927i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.326 + 1.37i)T \) |
| 3 | \( 1 + (1.15 + 2.76i)T \) |
| good | 5 | \( 1 + (-5.90 + 4.39i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-11.1 + 7.31i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.19 - 10.2i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (2.19 + 7.31i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (10.5 + 1.85i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (0.318 + 1.80i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (2.41 - 3.66i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (30.9 - 29.2i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (3.32 - 57.0i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-66.4 + 24.1i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-6.62 + 27.9i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-9.40 - 21.8i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-3.55 + 0.206i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (30.8 + 17.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-7.57 - 64.8i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (33.5 + 16.8i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-75.1 + 79.6i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (8.62 - 10.2i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (8.18 - 6.86i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (59.8 - 14.1i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-9.46 - 39.9i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-52.8 - 62.9i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (45.8 - 61.6i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−12.41727386791082437092839081524, −11.17535784687587878088759611137, −10.50057861216813690447360097846, −9.220439172257313849423489612289, −8.092516526695628797160804006152, −7.14814768938283022400487489343, −5.40567594241974421452450706323, −4.64919970834420183299530854962, −2.03884928164311072526808313344, −1.17389241289282368819180682580,
2.34320719033096176582071778893, 4.46379636304074076778348775251, 5.71613529030761588531473989329, 6.15590166418797946433314825891, 7.960081466624659160268822364600, 9.037452766776669541707836607744, 9.824397640091609219930344408877, 11.11218598045555379866249393700, 11.48307140467393382418409471673, 13.37340792839746838729364255501
