L(s) = 1 | − 3·3-s − 3·5-s − 4·7-s + 6·9-s − 5·11-s − 4·13-s + 9·15-s − 6·17-s − 7·19-s + 12·21-s + 3·23-s + 4·25-s − 9·27-s − 3·29-s − 4·31-s + 15·33-s + 12·35-s − 8·37-s + 12·39-s − 6·41-s − 8·43-s − 18·45-s − 6·47-s + 9·49-s + 18·51-s − 53-s + 15·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s − 1.50·11-s − 1.10·13-s + 2.32·15-s − 1.45·17-s − 1.60·19-s + 2.61·21-s + 0.625·23-s + 4/5·25-s − 1.73·27-s − 0.557·29-s − 0.718·31-s + 2.61·33-s + 2.02·35-s − 1.31·37-s + 1.92·39-s − 0.937·41-s − 1.21·43-s − 2.68·45-s − 0.875·47-s + 9/7·49-s + 2.52·51-s − 0.137·53-s + 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.99959877968722, −15.56115539228788, −15.36627443446851, −14.80978929640748, −13.58729608371114, −12.98123984188194, −12.69922070167028, −12.46947215282012, −11.66552156276745, −11.22512334930725, −10.79663081194918, −10.15984485865329, −9.936033696520783, −8.897069541842533, −8.395912241224259, −7.481703975526001, −6.998081870837675, −6.731992002723620, −6.015191499655305, −5.320239109892303, −4.733091673538975, −4.314934684182771, −3.475735290570383, −2.763855965713960, −1.806843103914617, 0, 0, 0,
1.806843103914617, 2.763855965713960, 3.475735290570383, 4.314934684182771, 4.733091673538975, 5.320239109892303, 6.015191499655305, 6.731992002723620, 6.998081870837675, 7.481703975526001, 8.395912241224259, 8.897069541842533, 9.936033696520783, 10.15984485865329, 10.79663081194918, 11.22512334930725, 11.66552156276745, 12.46947215282012, 12.69922070167028, 12.98123984188194, 13.58729608371114, 14.80978929640748, 15.36627443446851, 15.56115539228788, 15.99959877968722