| L(s) = 1 | − 4·3-s + 6·9-s + 12·17-s − 12·23-s − 25-s + 4·27-s + 12·29-s + 20·43-s + 10·49-s − 48·51-s − 12·53-s + 4·61-s + 48·69-s + 4·75-s + 16·79-s − 37·81-s − 48·87-s − 12·101-s − 28·103-s − 12·107-s − 12·113-s + 22·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 2.91·17-s − 2.50·23-s − 1/5·25-s + 0.769·27-s + 2.22·29-s + 3.04·43-s + 10/7·49-s − 6.72·51-s − 1.64·53-s + 0.512·61-s + 5.77·69-s + 0.461·75-s + 1.80·79-s − 4.11·81-s − 5.14·87-s − 1.19·101-s − 2.75·103-s − 1.16·107-s − 1.12·113-s + 2·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7956657465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7956657465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.782329130017813893019023675698, −8.105355048890860721915556903252, −8.102814624485965071133840858576, −7.70748000345931391286282647198, −7.28639401216852213802842988177, −6.66864946557653413060511748876, −6.41431444577913227147417397438, −6.08202396368087230331597442731, −5.72108161037187666210675518570, −5.46244489727843228818188171116, −5.33737447633867602657351279437, −4.63718705457336550358278868529, −4.33642021222658156372932472478, −3.86832379975189535173854779038, −3.35552926698506565891892195413, −2.70620006868094739265353761208, −2.37730660932038301990837985044, −1.23133730041348591495567225076, −1.09805004612256780152222336244, −0.40117057719229171308885344517,
0.40117057719229171308885344517, 1.09805004612256780152222336244, 1.23133730041348591495567225076, 2.37730660932038301990837985044, 2.70620006868094739265353761208, 3.35552926698506565891892195413, 3.86832379975189535173854779038, 4.33642021222658156372932472478, 4.63718705457336550358278868529, 5.33737447633867602657351279437, 5.46244489727843228818188171116, 5.72108161037187666210675518570, 6.08202396368087230331597442731, 6.41431444577913227147417397438, 6.66864946557653413060511748876, 7.28639401216852213802842988177, 7.70748000345931391286282647198, 8.102814624485965071133840858576, 8.105355048890860721915556903252, 8.782329130017813893019023675698
