| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 4·13-s + 16-s − 6·17-s + 18-s − 19-s − 6·23-s − 24-s − 5·25-s + 4·26-s − 27-s + 6·29-s − 2·31-s + 32-s − 6·34-s + 36-s − 4·37-s − 38-s − 4·39-s − 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s − 1.25·23-s − 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.657·37-s − 0.162·38-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.70397779901611905763392602462, −6.67685453158503036830492818980, −6.39461130745147042664210509886, −5.65473838318916067878138758246, −4.85679242074249456985197372563, −4.12049989296715617337624484159, −3.52845307997211445678021341580, −2.32322030951675246311285954744, −1.51207835751420731285626782871, 0,
1.51207835751420731285626782871, 2.32322030951675246311285954744, 3.52845307997211445678021341580, 4.12049989296715617337624484159, 4.85679242074249456985197372563, 5.65473838318916067878138758246, 6.39461130745147042664210509886, 6.67685453158503036830492818980, 7.70397779901611905763392602462
