| L(s) = 1 | + (−0.618 + 1.90i)2-s + (3.48 − 2.53i)3-s + (−3.23 − 2.35i)4-s + (−2.49 − 7.66i)5-s + (2.66 + 8.19i)6-s + (21.0 + 15.3i)7-s + (6.47 − 4.70i)8-s + (−2.60 + 8.02i)9-s + 16.1·10-s − 17.2·12-s + (−1.00 + 3.10i)13-s + (−42.1 + 30.6i)14-s + (−28.1 − 20.4i)15-s + (4.94 + 15.2i)16-s + (−6.44 − 19.8i)17-s + (−13.6 − 9.91i)18-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.670 − 0.487i)3-s + (−0.404 − 0.293i)4-s + (−0.222 − 0.685i)5-s + (0.181 + 0.557i)6-s + (1.13 + 0.827i)7-s + (0.286 − 0.207i)8-s + (−0.0965 + 0.297i)9-s + 0.509·10-s − 0.414·12-s + (−0.0215 + 0.0662i)13-s + (−0.805 + 0.585i)14-s + (−0.483 − 0.351i)15-s + (0.0772 + 0.237i)16-s + (−0.0920 − 0.283i)17-s + (−0.178 − 0.129i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.11338 + 0.310953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11338 + 0.310953i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 - 1.90i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (-3.48 + 2.53i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (2.49 + 7.66i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-21.0 - 15.3i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (1.00 - 3.10i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (6.44 + 19.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-101. + 74.0i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 - 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-213. - 155. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-61.6 + 189. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (295. + 214. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (221. - 160. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-41.9 + 30.4i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (127. - 392. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (21.2 + 15.4i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-50.7 - 156. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (159. + 491. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-195. - 141. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (84.3 - 259. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-22.4 - 69.0i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-452. + 1.39e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.81952653075352293230508786377, −10.80299808850363775177860249348, −9.189540636116537521348517597153, −8.668239933147688762124186209377, −7.896184847001484163667018286291, −7.00909160739507642190179137779, −5.35901789776722393195124450963, −4.75242945300469852121817538979, −2.68034655172719926596001637579, −1.18036045484269135614063586473,
1.20576974971142512780797088785, 2.94309421408336669865945261210, 3.83589917980480301124317244913, 5.00354640837594016577023633863, 6.86027142536089737561490041266, 7.945620843923277170962399649857, 8.712878956886131222886892333498, 9.938745274960742857793798625396, 10.56681988615405888884099592924, 11.47148463146932910820995650278
