| L(s) = 1 | + (0.618 + 1.90i)2-s + (−2.02 − 1.46i)3-s + (−3.23 + 2.35i)4-s + (1.54 − 4.75i)5-s + (1.54 − 4.75i)6-s + (−14.4 + 10.4i)7-s + (−6.47 − 4.70i)8-s + (−6.41 − 19.7i)9-s + 10.0·10-s + (−36.2 + 4.12i)11-s + 9.99·12-s + (−12.0 − 37.1i)13-s + (−28.8 − 20.9i)14-s + (−10.1 + 7.34i)15-s + (4.94 − 15.2i)16-s + (3.94 − 12.1i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.389 − 0.282i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.105 − 0.323i)6-s + (−0.777 + 0.565i)7-s + (−0.286 − 0.207i)8-s + (−0.237 − 0.731i)9-s + 0.316·10-s + (−0.993 + 0.113i)11-s + 0.240·12-s + (−0.257 − 0.792i)13-s + (−0.550 − 0.399i)14-s + (−0.174 + 0.126i)15-s + (0.0772 − 0.237i)16-s + (0.0562 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.846i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.174879 - 0.316321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174879 - 0.316321i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.618 - 1.90i)T \) |
| 5 | \( 1 + (-1.54 + 4.75i)T \) |
| 11 | \( 1 + (36.2 - 4.12i)T \) |
| good | 3 | \( 1 + (2.02 + 1.46i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + (14.4 - 10.4i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (12.0 + 37.1i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-3.94 + 12.1i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (95.3 + 69.2i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 62.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-11.0 + 8.03i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-49.7 - 153. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (182. - 132. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (213. + 155. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-126. - 92.0i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-54.3 - 167. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (701. - 509. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-95.0 + 292. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 732.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-203. + 625. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-976. + 709. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (252. + 778. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (53.8 - 165. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 892.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (445. + 1.37e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−12.69388453416269476373842427389, −12.32507239515825516412340926990, −10.69612926696695795792968019832, −9.376530640224620879369564155234, −8.428632926489387905859255218695, −7.00349815906581743971096265381, −5.97746766114428535195758478540, −4.96141183712938516336151297434, −3.00006574062740258294889728806, −0.18082874822178170785966694957,
2.37179088090313240999662750624, 3.94836016755950988878156257895, 5.33434993068681128842672018524, 6.65677820154571717266549949344, 8.159959547993356244860913398920, 9.736007700204023344231985294701, 10.50780834918155704274171032604, 11.20632009110183607615641926031, 12.54564138473846904785891021287, 13.41949613368273002538774977797
