L(s) = 1 | + (0.789 + 1.83i)2-s + (−3.65 + 2.11i)3-s + (−2.75 + 2.90i)4-s + (−1.06 − 1.84i)5-s + (−6.76 − 5.05i)6-s − 1.82i·7-s + (−7.50 − 2.77i)8-s + (4.42 − 7.65i)9-s + (2.55 − 3.42i)10-s + 13.8i·11-s + (3.94 − 16.4i)12-s + (−9.95 + 17.2i)13-s + (3.35 − 1.43i)14-s + (7.80 + 4.50i)15-s + (−0.826 − 15.9i)16-s + (3.83 + 6.63i)17-s + ⋯ |
L(s) = 1 | + (0.394 + 0.918i)2-s + (−1.21 + 0.703i)3-s + (−0.688 + 0.725i)4-s + (−0.213 − 0.369i)5-s + (−1.12 − 0.842i)6-s − 0.260i·7-s + (−0.938 − 0.346i)8-s + (0.491 − 0.850i)9-s + (0.255 − 0.342i)10-s + 1.25i·11-s + (0.329 − 1.36i)12-s + (−0.765 + 1.32i)13-s + (0.239 − 0.102i)14-s + (0.520 + 0.300i)15-s + (−0.0516 − 0.998i)16-s + (0.225 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0354277 - 0.693211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0354277 - 0.693211i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.789 - 1.83i)T \) |
| 19 | \( 1 + (-16.7 - 9.04i)T \) |
good | 3 | \( 1 + (3.65 - 2.11i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (1.06 + 1.84i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + 1.82iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 + (9.95 - 17.2i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-3.83 - 6.63i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (9.14 + 5.28i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (0.0147 - 0.0255i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 42.2iT - 961T^{2} \) |
| 37 | \( 1 - 19.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-15.7 - 27.2i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-51.3 + 29.6i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (77.5 + 44.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (29.8 - 51.6i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (63.9 - 36.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (23.6 - 40.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (5.77 + 3.33i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-24.1 + 13.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (46.2 + 80.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-110. + 63.9i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 88.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-31.2 + 54.0i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-64.8 - 112. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−14.97647035420843670441857189193, −14.00330277398577114135265219561, −12.35836178788841053853773446117, −11.95182031590234516434793789851, −10.29715474131631292041456309199, −9.254893442040718110577892424534, −7.55553039062788413703664465749, −6.38720109819317263962293395968, −4.97646083540642332458945232243, −4.28271658090919043225218375051,
0.61652854025098143434612788386, 3.05559685728940352279397694561, 5.22887431042082597593092903205, 6.06264225932595380499310549915, 7.73110331607055842060066230865, 9.549203271603088020153348395288, 10.95376663894320842989992755462, 11.44560668889554689408068939362, 12.47055037320261131076155899815, 13.29129349193259507635720231460