L(s) = 1 | + 1.81i·2-s + (−3.99 + 3.31i)3-s + 4.69·4-s + (−5.16 + 8.94i)5-s + (−6.03 − 7.26i)6-s + (−16.8 − 7.78i)7-s + 23.0i·8-s + (4.97 − 26.5i)9-s + (−16.2 − 9.38i)10-s + (−10.1 + 5.87i)11-s + (−18.7 + 15.5i)12-s + (−46.7 + 26.9i)13-s + (14.1 − 30.5i)14-s + (−9.03 − 52.8i)15-s − 4.40·16-s + (−31.9 + 55.3i)17-s + ⋯ |
L(s) = 1 | + 0.642i·2-s + (−0.769 + 0.638i)3-s + 0.586·4-s + (−0.461 + 0.799i)5-s + (−0.410 − 0.494i)6-s + (−0.907 − 0.420i)7-s + 1.01i·8-s + (0.184 − 0.982i)9-s + (−0.514 − 0.296i)10-s + (−0.278 + 0.161i)11-s + (−0.451 + 0.374i)12-s + (−0.997 + 0.575i)13-s + (0.270 − 0.583i)14-s + (−0.155 − 0.910i)15-s − 0.0687·16-s + (−0.456 + 0.790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0612197 + 0.852258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0612197 + 0.852258i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.99 - 3.31i)T \) |
| 7 | \( 1 + (16.8 + 7.78i)T \) |
good | 2 | \( 1 - 1.81iT - 8T^{2} \) |
| 5 | \( 1 + (5.16 - 8.94i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (10.1 - 5.87i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (46.7 - 26.9i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (31.9 - 55.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-87.6 + 50.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-168. - 97.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-13.4 - 7.77i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 200. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (152. + 264. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-35.3 - 61.2i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-52.4 + 90.9i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 7.27T + 1.03e5T^{2} \) |
| 53 | \( 1 + (460. + 266. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 301. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 709. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-732. - 423. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-164. + 284. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (506. + 876. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.20e3 - 694. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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
−15.36279158549561761833389867616, −14.32610579993096523001641233054, −12.59445626429968640263559806220, −11.36491420444875467789368807117, −10.64878246845340976315020604733, −9.367122124211841215633743976986, −7.23244262580690612694610292552, −6.72956062679158136553437111380, −5.17944822762510927177974286360, −3.26562097220768953198208952668,
0.62277209056304601931926144861, 2.76615596741089459113831463133, 5.06641288477064207311053202689, 6.56217270100151630980277835039, 7.74305347339055230003556451838, 9.531604354559304189353170597854, 10.77075651205512751009547283286, 11.98628311063830073529623572468, 12.43338169361439416703854554323, 13.34170727213200135512642307105