| L(s) = 1 | + (−0.697 − 1.87i)2-s + (0.927 − 3.46i)3-s + (−3.02 + 2.61i)4-s + (−1.39 − 4.80i)5-s + (−7.13 + 0.675i)6-s + (−6.90 + 1.16i)7-s + (7.01 + 3.85i)8-s + (−3.32 − 1.91i)9-s + (−8.02 + 5.96i)10-s + (−7.98 + 4.61i)11-s + (6.24 + 12.9i)12-s + (12.7 − 12.7i)13-s + (6.99 + 12.1i)14-s + (−17.9 + 0.368i)15-s + (2.32 − 15.8i)16-s + (−0.0413 + 0.154i)17-s + ⋯ |
| L(s) = 1 | + (−0.348 − 0.937i)2-s + (0.309 − 1.15i)3-s + (−0.756 + 0.653i)4-s + (−0.278 − 0.960i)5-s + (−1.18 + 0.112i)6-s + (−0.986 + 0.166i)7-s + (0.876 + 0.481i)8-s + (−0.369 − 0.213i)9-s + (−0.802 + 0.596i)10-s + (−0.726 + 0.419i)11-s + (0.520 + 1.07i)12-s + (0.977 − 0.977i)13-s + (0.499 + 0.866i)14-s + (−1.19 + 0.0245i)15-s + (0.145 − 0.989i)16-s + (−0.00243 + 0.00907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.536i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.223272 + 0.767865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223272 + 0.767865i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.697 + 1.87i)T \) |
| 5 | \( 1 + (1.39 + 4.80i)T \) |
| 7 | \( 1 + (6.90 - 1.16i)T \) |
| good | 3 | \( 1 + (-0.927 + 3.46i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (7.98 - 4.61i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-12.7 + 12.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (0.0413 - 0.154i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (16.3 + 9.43i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.72 - 28.8i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 12.2iT - 841T^{2} \) |
| 31 | \( 1 + (23.7 + 41.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (8.80 + 32.8i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 5.88iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-32.4 + 32.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (10.3 + 38.5i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-17.5 + 65.3i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-12.6 + 7.29i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-58.5 - 33.7i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (18.5 - 69.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 38.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (33.0 + 8.85i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-51.1 + 88.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-57.8 - 57.8i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (74.9 - 129. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-5.13 - 5.13i)T + 9.40e3iT^{2} \) |
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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
−12.64167421420126205498118501813, −11.51763580862192896826784082787, −10.24488448932454025976222056105, −9.104448394502913382128541946939, −8.203188726348695610904294435522, −7.31051084738106841070276595738, −5.55253546222648551704117699615, −3.78202480315148708491876850136, −2.21827467776685167108723146240, −0.57528978879417585944188722673,
3.34481638303580909880419086863, 4.43093883949048326058250284936, 6.13127239276350670913664405018, 6.95330932903818293010485520393, 8.433210119431640056112354186566, 9.298819472559397401287685234659, 10.43528239369476021757488365922, 10.78090090673753106504027372902, 12.78022195429232354737799363647, 13.98279267219759028543213180036
