| L(s) = 1 | + 4.57i·3-s + 209. i·7-s + 222.·9-s − 121·11-s + 100. i·13-s + 978. i·17-s − 1.35e3·19-s − 957.·21-s + 2.07e3i·23-s + 2.12e3i·27-s + 4.87e3·29-s − 6.30e3·31-s − 553. i·33-s − 541. i·37-s − 460.·39-s + ⋯ |
| L(s) = 1 | + 0.293i·3-s + 1.61i·7-s + 0.913·9-s − 0.301·11-s + 0.165i·13-s + 0.821i·17-s − 0.859·19-s − 0.474·21-s + 0.818i·23-s + 0.562i·27-s + 1.07·29-s − 1.17·31-s − 0.0885i·33-s − 0.0650i·37-s − 0.0484·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.116588954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116588954\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 4.57iT - 243T^{2} \) |
| 7 | \( 1 - 209. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 100. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 978. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.07e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 541. iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.10e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.78e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 7.75e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.65e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.28e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.33e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.17e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.30e4iT - 8.58e9T^{2} \) |
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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
−9.541177384378772181190053386849, −8.808493736649738691112404048374, −8.145843133349777116901772857597, −7.06373372430142317374029304125, −6.13299368490930207642538136159, −5.38228253229574639388389998816, −4.48475517861725488588317987357, −3.45371093492035312334919108288, −2.32028000850917794412946737330, −1.51109798408930275191290736642,
0.21958748931528972100012494695, 1.00399527498375262880513036906, 2.10655300103796123117836750873, 3.42500557979857258995416102105, 4.31498130189266359209343360472, 4.99899325181694724794029351008, 6.45008530668366185101137008872, 7.01382528753420104195893889373, 7.66666875332713262056129542470, 8.520249458058456533699302828888
