| L(s) = 1 | − 3·3-s + 5·5-s − 8·7-s + 9·9-s − 20·11-s − 34·13-s − 15·15-s − 6·17-s + 12·19-s + 24·21-s + 120·23-s + 25·25-s − 27·27-s + 246·29-s + 31·31-s + 60·33-s − 40·35-s + 310·37-s + 102·39-s − 518·41-s + 92·43-s + 45·45-s − 88·47-s − 279·49-s + 18·51-s − 738·53-s − 100·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.431·7-s + 1/3·9-s − 0.548·11-s − 0.725·13-s − 0.258·15-s − 0.0856·17-s + 0.144·19-s + 0.249·21-s + 1.08·23-s + 1/5·25-s − 0.192·27-s + 1.57·29-s + 0.179·31-s + 0.316·33-s − 0.193·35-s + 1.37·37-s + 0.418·39-s − 1.97·41-s + 0.326·43-s + 0.149·45-s − 0.273·47-s − 0.813·49-s + 0.0494·51-s − 1.91·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 31 | \( 1 - p T \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 12 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 37 | \( 1 - 310 T + p^{3} T^{2} \) |
| 41 | \( 1 + 518 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 88 T + p^{3} T^{2} \) |
| 53 | \( 1 + 738 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 220 T + p^{3} T^{2} \) |
| 71 | \( 1 + 512 T + p^{3} T^{2} \) |
| 73 | \( 1 + 758 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 18 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1634 T + p^{3} T^{2} \) |
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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
−8.495893720630614149658607882914, −7.62087170026264427934048737349, −6.72658903580713775022663854163, −6.19899721891415180092922270877, −5.10796763704213823547994109447, −4.70154192755217439390450092362, −3.28742937552686409862019983166, −2.45089327650254931770986989090, −1.15576524145047880151924702834, 0,
1.15576524145047880151924702834, 2.45089327650254931770986989090, 3.28742937552686409862019983166, 4.70154192755217439390450092362, 5.10796763704213823547994109447, 6.19899721891415180092922270877, 6.72658903580713775022663854163, 7.62087170026264427934048737349, 8.495893720630614149658607882914
