| L(s) = 1 | + (1.23 − 3.80i)2-s + (−0.518 − 1.59i)3-s + (−12.9 − 9.40i)4-s + 52.9·5-s − 6.71·6-s + (102. + 74.4i)7-s + (−51.7 + 37.6i)8-s + (194. − 141. i)9-s + (65.4 − 201. i)10-s + (−128. − 93.0i)11-s + (−8.29 + 25.5i)12-s + (−182. − 562. i)13-s + (409. − 297. i)14-s + (−27.4 − 84.5i)15-s + (79.1 + 243. i)16-s + (949. − 689. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.0332 − 0.102i)3-s + (−0.404 − 0.293i)4-s + 0.947·5-s − 0.0761·6-s + (0.789 + 0.573i)7-s + (−0.286 + 0.207i)8-s + (0.799 − 0.580i)9-s + (0.207 − 0.637i)10-s + (−0.319 − 0.231i)11-s + (−0.0166 + 0.0512i)12-s + (−0.299 − 0.922i)13-s + (0.558 − 0.405i)14-s + (−0.0315 − 0.0970i)15-s + (0.0772 + 0.237i)16-s + (0.796 − 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.81060 - 1.42219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81060 - 1.42219i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-4.87e3 - 2.21e3i)T \) |
| good | 3 | \( 1 + (0.518 + 1.59i)T + (-196. + 142. i)T^{2} \) |
| 5 | \( 1 - 52.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + (-102. - 74.4i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 11 | \( 1 + (128. + 93.0i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (182. + 562. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-949. + 689. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-149. + 460. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (170. - 123. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-705. + 2.17e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 37 | \( 1 - 3.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (5.53e3 - 1.70e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + (283. - 872. i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-8.25e3 - 2.54e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.19e4 - 8.66e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (301. + 926. i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.05e4 - 7.67e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.18e4 - 2.31e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.08e4 - 1.51e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.35e4 + 7.24e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (9.61e4 + 6.98e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.56e4 - 1.13e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.61222943217356414662221922803, −12.63239162817979785139295750961, −11.62829165201019857445974867145, −10.25629424939150607109271729142, −9.438082441841192159113851453892, −7.904680512862872813421526177135, −6.02324668257736925405935415555, −4.84689325247949302863161309569, −2.79277269955240885425557282182, −1.22194860961717575759486100318,
1.75956432691718057523309992259, 4.28130309226861170025909181718, 5.46567601905514123445462443938, 6.97269955803452046783910782158, 8.080066063671189078413375176165, 9.660997278058188608701899325758, 10.58266787625315528573557356026, 12.22731757569619042411185581352, 13.53328014955128074806333257112, 14.11388018518049487755491376888
