| L(s) = 1 | + (1.23 + 3.80i)2-s + (7.45 − 22.9i)3-s + (−12.9 + 9.40i)4-s + 91.0·5-s + 96.5·6-s + (−63.5 + 46.1i)7-s + (−51.7 − 37.6i)8-s + (−274. − 199. i)9-s + (112. + 346. i)10-s + (265. − 192. i)11-s + (119. + 367. i)12-s + (295. − 909. i)13-s + (−254. − 184. i)14-s + (679. − 2.08e3i)15-s + (79.1 − 243. i)16-s + (117. + 85.3i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.478 − 1.47i)3-s + (−0.404 + 0.293i)4-s + 1.62·5-s + 1.09·6-s + (−0.490 + 0.356i)7-s + (−0.286 − 0.207i)8-s + (−1.13 − 0.821i)9-s + (0.355 + 1.09i)10-s + (0.660 − 0.480i)11-s + (0.239 + 0.736i)12-s + (0.484 − 1.49i)13-s + (−0.346 − 0.251i)14-s + (0.779 − 2.39i)15-s + (0.0772 − 0.237i)16-s + (0.0985 + 0.0716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.51952 - 0.790192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51952 - 0.790192i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 3.80i)T \) |
| 31 | \( 1 + (-311. - 5.34e3i)T \) |
| good | 3 | \( 1 + (-7.45 + 22.9i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 - 91.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + (63.5 - 46.1i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (-265. + 192. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-295. + 909. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-117. - 85.3i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-631. - 1.94e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.67e3 + 1.94e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (429. + 1.32e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 37 | \( 1 - 3.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (2.73e3 + 8.40e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + (-3.51e3 - 1.08e4i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (7.95e3 - 2.44e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-889. - 646. i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.30e4 - 4.01e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + 2.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.97e4 - 4.33e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.81e4 - 3.49e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-3.23e4 - 2.35e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.47e4 + 4.54e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-9.57e4 + 6.95e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.01e5 + 7.40e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.90990409648371735979595207613, −12.99909828706732330339684113161, −12.35138637069400873322369186317, −10.17305441912344960348300577068, −8.874277076972679964260737537800, −7.80135940865357379311131988001, −6.21083667568343696144244187263, −5.91453210170703162533785235671, −2.91612304594794761679435070588, −1.30404499309612954080189682698,
1.99241900447971829279542833990, 3.65607996090391742403428792891, 4.89452469808051385613545017608, 6.42060772282767087090581347945, 9.207489776429562123728898606888, 9.461054562137625186020948078862, 10.35611981357725431894618154501, 11.60735565369585713011119340325, 13.43488930495987575150663123898, 13.93179757119033467978793965674
