| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 7-s + 8-s + 9-s + 10-s + 2·11-s − 2·12-s + 3·13-s + 14-s − 2·15-s + 16-s − 6·17-s + 18-s + 5·19-s + 20-s − 2·21-s + 2·22-s + 2·23-s − 2·24-s + 25-s + 3·26-s + 4·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.577·12-s + 0.832·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.436·21-s + 0.426·22-s + 0.417·23-s − 0.408·24-s + 1/5·25-s + 0.588·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.92421552463152, −14.49180997430685, −13.95045836356837, −13.36790796762331, −13.07823981227104, −12.33590428304313, −11.91311525022300, −11.27497160819582, −11.02992571748501, −10.75182804925273, −9.735457589680533, −9.405331533125579, −8.571629169961494, −8.157035561565178, −7.141266548368313, −6.625472893964740, −6.472649848808096, −5.536184657909442, −5.276799810025468, −4.802241606672229, −3.920386039266316, −3.475208548144288, −2.529993680048421, −1.702829270088732, −1.127034741594633, 0,
1.127034741594633, 1.702829270088732, 2.529993680048421, 3.475208548144288, 3.920386039266316, 4.802241606672229, 5.276799810025468, 5.536184657909442, 6.472649848808096, 6.625472893964740, 7.141266548368313, 8.157035561565178, 8.571629169961494, 9.405331533125579, 9.735457589680533, 10.75182804925273, 11.02992571748501, 11.27497160819582, 11.91311525022300, 12.33590428304313, 13.07823981227104, 13.36790796762331, 13.95045836356837, 14.49180997430685, 14.92421552463152
