| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s − 2·9-s + 10-s + 2·11-s + 12-s + 3·13-s + 14-s + 15-s + 16-s + 3·17-s − 2·18-s − 7·19-s + 20-s + 21-s + 2·22-s − 4·23-s + 24-s + 25-s + 3·26-s − 5·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 1.60·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.962·27-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 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.75150174489820, −14.53259957830740, −14.10375640279284, −13.52347385542639, −13.09074886544800, −12.57822060543052, −11.90336637771724, −11.45441209196692, −10.94921530538569, −10.37801517573757, −9.801928919015963, −9.065629170034606, −8.666010661551682, −8.076363756317478, −7.664176593058672, −6.731792720168314, −6.193886071341898, −5.911060236832628, −5.130688279355438, −4.499031016109268, −3.777644124599970, −3.375285361980626, −2.597247182149340, −1.871255391523782, −1.407985510399239, 0,
1.407985510399239, 1.871255391523782, 2.597247182149340, 3.375285361980626, 3.777644124599970, 4.499031016109268, 5.130688279355438, 5.911060236832628, 6.193886071341898, 6.731792720168314, 7.664176593058672, 8.076363756317478, 8.666010661551682, 9.065629170034606, 9.801928919015963, 10.37801517573757, 10.94921530538569, 11.45441209196692, 11.90336637771724, 12.57822060543052, 13.09074886544800, 13.52347385542639, 14.10375640279284, 14.53259957830740, 14.75150174489820
