L(s) = 1 | + (4.36 − 2.52i)2-s + (−1.5 − 2.59i)3-s + (8.72 − 15.1i)4-s + 20.1i·5-s + (−13.1 − 7.56i)6-s + (−13.3 − 7.71i)7-s − 47.7i·8-s + (−4.5 + 7.79i)9-s + (50.7 + 87.9i)10-s + (23.3 − 13.4i)11-s − 52.3·12-s + (−3.96 + 46.7i)13-s − 77.8·14-s + (52.2 − 30.1i)15-s + (−50.5 − 87.5i)16-s + (11.6 − 20.1i)17-s + ⋯ |
L(s) = 1 | + (1.54 − 0.891i)2-s + (−0.288 − 0.499i)3-s + (1.09 − 1.88i)4-s + 1.79i·5-s + (−0.891 − 0.514i)6-s + (−0.721 − 0.416i)7-s − 2.10i·8-s + (−0.166 + 0.288i)9-s + (1.60 + 2.77i)10-s + (0.639 − 0.369i)11-s − 1.25·12-s + (−0.0845 + 0.996i)13-s − 1.48·14-s + (0.899 − 0.519i)15-s + (−0.789 − 1.36i)16-s + (0.165 − 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.00732 - 1.29319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00732 - 1.29319i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 13 | \( 1 + (3.96 - 46.7i)T \) |
good | 2 | \( 1 + (-4.36 + 2.52i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 20.1iT - 125T^{2} \) |
| 7 | \( 1 + (13.3 + 7.71i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-23.3 + 13.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-11.6 + 20.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (39.0 + 22.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (71.0 + 122. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (1.14 + 1.98i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 37.7iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-271. + 156. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-5.08 + 2.93i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (180. - 312. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 209. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-470. - 271. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (102. - 178. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (426. - 246. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-716. - 413. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 - 66.1iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-555. + 320. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (965. + 557. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.79986980376336269334352413802, −14.18822128832894418498280125440, −13.25809200816564607948788017369, −11.88450504717024711540831562530, −11.09515614652828434341156376399, −10.07724789007631260842628684216, −6.86026043971086118409834433336, −6.22692635749115362803275122280, −3.92728289114181318022981366204, −2.52517723789855038838594333263,
3.87908371896794475503951386736, 5.13359945187867324147800391272, 6.08254506186956173159111194341, 8.070835238840868105464823483322, 9.546132345129588413935584009692, 11.93972869815489836688473260411, 12.61387347115097425196707142210, 13.45510914175025512135770635156, 15.03167051457534373631140113294, 15.81460450956783761617889357571