L(s) = 1 | + 3i·3-s + (8.10 + 7.70i)5-s + 22.2i·7-s − 9·9-s − 1.79·11-s − 58.2i·13-s + (−23.1 + 24.3i)15-s − 18.9i·17-s − 104.·19-s − 66.6·21-s + 49.6i·23-s + (6.37 + 124. i)25-s − 27i·27-s − 293.·29-s − 64.4·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.724 + 0.688i)5-s + 1.19i·7-s − 0.333·9-s − 0.0490·11-s − 1.24i·13-s + (−0.397 + 0.418i)15-s − 0.270i·17-s − 1.26·19-s − 0.692·21-s + 0.449i·23-s + (0.0509 + 0.998i)25-s − 0.192i·27-s − 1.87·29-s − 0.373·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.724i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.688 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4531987337\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4531987337\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (-8.10 - 7.70i)T \) |
good | 7 | \( 1 - 22.2iT - 343T^{2} \) |
| 11 | \( 1 + 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 18.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 19.8iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 247. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 384. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 463. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 173. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 320. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 173. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 384. iT - 9.12e5T^{2} \) |
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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
−10.13163772418681391627659010430, −9.389641359705812230403928354379, −8.696048629718463407870908186203, −7.73023831391309734830218666382, −6.60169732000237952391599215220, −5.63703641658160895088353320896, −5.30778269588247831751305868736, −3.77553096644257448858535320743, −2.80073767316836507443678076121, −1.95616523775568600498664745670,
0.10466918239197473057441049062, 1.39653221423124725637504785611, 2.17651159235255094976936830490, 3.85553622034528379319277810257, 4.59883035664238473308175217030, 5.77135411216349948249839241662, 6.62612940830560114252244471088, 7.30067402666960912461670581748, 8.349699690920884724558978473285, 9.061399834267445620381828865369