| L(s) = 1 | − 3·3-s + 16·5-s + 7·7-s + 9·9-s + 18·11-s + 54·13-s − 48·15-s − 128·17-s − 52·19-s − 21·21-s − 202·23-s + 131·25-s − 27·27-s − 302·29-s − 200·31-s − 54·33-s + 112·35-s + 150·37-s − 162·39-s + 172·41-s − 164·43-s + 144·45-s − 460·47-s + 49·49-s + 384·51-s + 190·53-s + 288·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.43·5-s + 0.377·7-s + 1/3·9-s + 0.493·11-s + 1.15·13-s − 0.826·15-s − 1.82·17-s − 0.627·19-s − 0.218·21-s − 1.83·23-s + 1.04·25-s − 0.192·27-s − 1.93·29-s − 1.15·31-s − 0.284·33-s + 0.540·35-s + 0.666·37-s − 0.665·39-s + 0.655·41-s − 0.581·43-s + 0.477·45-s − 1.42·47-s + 1/7·49-s + 1.05·51-s + 0.492·53-s + 0.706·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 128 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 202 T + p^{3} T^{2} \) |
| 29 | \( 1 + 302 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 172 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 460 T + p^{3} T^{2} \) |
| 53 | \( 1 - 190 T + p^{3} T^{2} \) |
| 59 | \( 1 + 96 T + p^{3} T^{2} \) |
| 61 | \( 1 + 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 744 T + p^{3} T^{2} \) |
| 71 | \( 1 + 54 T + p^{3} T^{2} \) |
| 73 | \( 1 - 742 T + p^{3} T^{2} \) |
| 79 | \( 1 + 92 T + p^{3} T^{2} \) |
| 83 | \( 1 - 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 116 T + p^{3} T^{2} \) |
| 97 | \( 1 + 554 T + p^{3} T^{2} \) |
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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
−9.072067827652421736942438678734, −8.119451438195215093783255450393, −6.93458875499101118425116741666, −6.05925839063201464935725462723, −5.85981919890935199594940688154, −4.62338667788896377601597435062, −3.78620963024430969694954691990, −2.07000054283250989389776010577, −1.65376857011771096760186628285, 0,
1.65376857011771096760186628285, 2.07000054283250989389776010577, 3.78620963024430969694954691990, 4.62338667788896377601597435062, 5.85981919890935199594940688154, 6.05925839063201464935725462723, 6.93458875499101118425116741666, 8.119451438195215093783255450393, 9.072067827652421736942438678734
