| L(s) = 1 | + 3-s + 5-s − 5·7-s − 2·9-s + 2·11-s + 4·13-s + 15-s + 17-s + 7·19-s − 5·21-s − 2·23-s − 4·25-s − 5·27-s + 3·29-s − 4·31-s + 2·33-s − 5·35-s + 8·37-s + 4·39-s + 5·41-s − 4·43-s − 2·45-s − 8·47-s + 18·49-s + 51-s − 53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.88·7-s − 2/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s + 0.242·17-s + 1.60·19-s − 1.09·21-s − 0.417·23-s − 4/5·25-s − 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.348·33-s − 0.845·35-s + 1.31·37-s + 0.640·39-s + 0.780·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s + 18/7·49-s + 0.140·51-s − 0.137·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.34603207102447, −13.83100820942305, −13.61675303455495, −13.07466957742647, −12.64917751392987, −11.93153470666588, −11.54043084623126, −10.99289071595201, −10.17209754857371, −9.756194914586419, −9.427826087829989, −9.001376506002045, −8.442923613066118, −7.728570612489524, −7.284440590190743, −6.462779567546115, −6.044069759950492, −5.865022446575955, −5.039963950571819, −3.980347495052896, −3.629110092449210, −3.060297225504129, −2.692567126230968, −1.721637597355946, −0.9472475273207676, 0,
0.9472475273207676, 1.721637597355946, 2.692567126230968, 3.060297225504129, 3.629110092449210, 3.980347495052896, 5.039963950571819, 5.865022446575955, 6.044069759950492, 6.462779567546115, 7.284440590190743, 7.728570612489524, 8.442923613066118, 9.001376506002045, 9.427826087829989, 9.756194914586419, 10.17209754857371, 10.99289071595201, 11.54043084623126, 11.93153470666588, 12.64917751392987, 13.07466957742647, 13.61675303455495, 13.83100820942305, 14.34603207102447
