| L(s) = 1 | + 2·3-s − 5-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 17-s + 19-s + 23-s − 4·25-s − 4·27-s − 6·29-s + 4·31-s + 8·33-s − 3·37-s + 4·39-s − 10·41-s − 43-s − 45-s + 4·47-s − 2·51-s + 6·53-s − 4·55-s + 2·57-s + 59-s − 10·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.242·17-s + 0.229·19-s + 0.208·23-s − 4/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s + 1.39·33-s − 0.493·37-s + 0.640·39-s − 1.56·41-s − 0.152·43-s − 0.149·45-s + 0.583·47-s − 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.264·57-s + 0.130·59-s − 1.28·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.26087567267904, −15.09897363708665, −14.53867798747890, −13.87758666154384, −13.54684833366219, −13.17996009351847, −12.06091826311547, −12.00508753008100, −11.29191529965113, −10.72202155929722, −9.963857411815410, −9.345098657065708, −8.997182332903326, −8.421661368925970, −7.975725711518288, −7.300079307799023, −6.772707724500400, −6.059412017128804, −5.413384897732091, −4.476358124338526, −3.829939645494582, −3.518994489528510, −2.765853531467547, −1.906579633195407, −1.284041603218250, 0,
1.284041603218250, 1.906579633195407, 2.765853531467547, 3.518994489528510, 3.829939645494582, 4.476358124338526, 5.413384897732091, 6.059412017128804, 6.772707724500400, 7.300079307799023, 7.975725711518288, 8.421661368925970, 8.997182332903326, 9.345098657065708, 9.963857411815410, 10.72202155929722, 11.29191529965113, 12.00508753008100, 12.06091826311547, 13.17996009351847, 13.54684833366219, 13.87758666154384, 14.53867798747890, 15.09897363708665, 15.26087567267904
