| L(s) = 1 | − 2·3-s − i·5-s − 2i·7-s + 9-s + 2i·15-s + 6·17-s − 4i·19-s + 4i·21-s − 6·23-s − 25-s + 4·27-s + 6·29-s − 4i·31-s − 2·35-s − 2i·37-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447i·5-s − 0.755i·7-s + 0.333·9-s + 0.516i·15-s + 1.45·17-s − 0.917i·19-s + 0.872i·21-s − 1.25·23-s − 0.200·25-s + 0.769·27-s + 1.11·29-s − 0.718i·31-s − 0.338·35-s − 0.328i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8920009790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8920009790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.102814624485965071133840858576, −7.70748000345931391286282647198, −6.66864946557653413060511748876, −6.08202396368087230331597442731, −5.33737447633867602657351279437, −4.63718705457336550358278868529, −3.86832379975189535173854779038, −2.70620006868094739265353761208, −1.23133730041348591495567225076, −0.40117057719229171308885344517,
1.09805004612256780152222336244, 2.37730660932038301990837985044, 3.35552926698506565891892195413, 4.33642021222658156372932472478, 5.46244489727843228818188171116, 5.72108161037187666210675518570, 6.41431444577913227147417397438, 7.28639401216852213802842988177, 8.105355048890860721915556903252, 8.782329130017813893019023675698
