L(s) = 1 | + (−0.112 − 0.195i)2-s + (−2.06 − 4.76i)3-s + (3.97 − 6.88i)4-s + (−0.697 + 0.940i)6-s + (−15.5 − 27.0i)7-s − 3.59·8-s + (−18.4 + 19.7i)9-s + (−9.06 − 15.6i)11-s + (−41.0 − 4.71i)12-s + (−25.0 + 43.4i)13-s + (−3.51 + 6.08i)14-s + (−31.3 − 54.3i)16-s + 131.·17-s + (5.92 + 1.37i)18-s + 23.2·19-s + ⋯ |
L(s) = 1 | + (−0.0398 − 0.0689i)2-s + (−0.397 − 0.917i)3-s + (0.496 − 0.860i)4-s + (−0.0474 + 0.0639i)6-s + (−0.842 − 1.45i)7-s − 0.158·8-s + (−0.683 + 0.730i)9-s + (−0.248 − 0.430i)11-s + (−0.987 − 0.113i)12-s + (−0.535 + 0.926i)13-s + (−0.0670 + 0.116i)14-s + (−0.490 − 0.849i)16-s + 1.87·17-s + (0.0775 + 0.0180i)18-s + 0.280·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.246364 + 0.828893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.246364 + 0.828893i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.06 + 4.76i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.112 + 0.195i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (15.5 + 27.0i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (9.06 + 15.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (25.0 - 43.4i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (16.4 - 28.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (62.9 + 108. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (62.5 - 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 99.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + (122. - 212. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-69.5 - 120. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (236. + 409. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 421.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-371. + 642. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (4.48 + 7.77i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-294. + 510. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 48.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 409.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (265. + 459. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (147. + 255. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 852.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (194. + 336. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.26253549655335775302633540093, −10.28098206056054409700995679416, −9.633626570328707259968291779205, −7.82845398708889823929794301566, −7.03778075061073782829134848368, −6.25945539000631164637649471338, −5.13606688145294082517484706489, −3.29516059918888976725107807043, −1.55333102184869040779465892175, −0.37168556973712492837050373942,
2.71060414474042798112181051567, 3.50923362047646336743960850692, 5.26588912573304506309185538560, 5.99882965244231894235456386608, 7.41040121098156426756732751535, 8.535437589107960922458506707621, 9.539536753829973701789407502445, 10.31452973756510643464569787631, 11.58627255747310594845439577947, 12.36181499252677672854805440703