L(s) = 1 | − 2·3-s + 9-s − 3·11-s − 5·13-s − 6·17-s + 19-s + 3·23-s + 4·27-s − 6·29-s + 4·31-s + 6·33-s − 11·37-s + 10·39-s + 3·41-s − 10·43-s + 3·47-s + 12·51-s − 3·53-s − 2·57-s − 4·61-s − 4·67-s − 6·69-s − 12·71-s + 4·73-s + 10·79-s − 11·81-s − 12·83-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.904·11-s − 1.38·13-s − 1.45·17-s + 0.229·19-s + 0.625·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 1.04·33-s − 1.80·37-s + 1.60·39-s + 0.468·41-s − 1.52·43-s + 0.437·47-s + 1.68·51-s − 0.412·53-s − 0.264·57-s − 0.512·61-s − 0.488·67-s − 0.722·69-s − 1.42·71-s + 0.468·73-s + 1.12·79-s − 1.22·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.31447028886438, −15.72467624010475, −15.20635532680149, −14.79204192289731, −13.94604424525803, −13.41408916246425, −12.87993725255524, −12.28682287710811, −11.82984405186860, −11.30731035774524, −10.62099796763905, −10.43950985477718, −9.593378966877386, −9.044704004534090, −8.325166952016229, −7.630079442485081, −6.931281174874552, −6.630429844834309, −5.728803054385514, −5.150674354017785, −4.870137471995982, −4.080887599569857, −2.990981017356400, −2.420595793613424, −1.466367647013218, 0, 0,
1.466367647013218, 2.420595793613424, 2.990981017356400, 4.080887599569857, 4.870137471995982, 5.150674354017785, 5.728803054385514, 6.630429844834309, 6.931281174874552, 7.630079442485081, 8.325166952016229, 9.044704004534090, 9.593378966877386, 10.43950985477718, 10.62099796763905, 11.30731035774524, 11.82984405186860, 12.28682287710811, 12.87993725255524, 13.41408916246425, 13.94604424525803, 14.79204192289731, 15.20635532680149, 15.72467624010475, 16.31447028886438