| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s − 4·13-s − 4·17-s − 4·19-s + 4·21-s + 8·23-s − 27-s + 6·29-s + 4·31-s − 4·33-s − 2·37-s + 4·39-s + 10·41-s − 43-s + 4·47-s + 9·49-s + 4·51-s + 2·53-s + 4·57-s + 12·59-s − 4·63-s + 8·67-s − 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.970·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.152·43-s + 0.583·47-s + 9/7·49-s + 0.560·51-s + 0.274·53-s + 0.529·57-s + 1.56·59-s − 0.503·63-s + 0.977·67-s − 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356982529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356982529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.57054136092293, −13.93288372952460, −13.33705976969554, −12.82809144832201, −12.46122807877989, −12.08578135012954, −11.35685138291936, −10.95190700773856, −10.31434702843558, −9.795555698932940, −9.357712839189096, −8.903604434430710, −8.331590205018283, −7.356632650329248, −6.848211557769570, −6.548492768756733, −6.201389044783886, −5.289644030534003, −4.769745522857566, −4.084660855438559, −3.628952870839591, −2.638958701078731, −2.373610032129625, −1.087568689194519, −0.4887947982332246,
0.4887947982332246, 1.087568689194519, 2.373610032129625, 2.638958701078731, 3.628952870839591, 4.084660855438559, 4.769745522857566, 5.289644030534003, 6.201389044783886, 6.548492768756733, 6.848211557769570, 7.356632650329248, 8.331590205018283, 8.903604434430710, 9.357712839189096, 9.795555698932940, 10.31434702843558, 10.95190700773856, 11.35685138291936, 12.08578135012954, 12.46122807877989, 12.82809144832201, 13.33705976969554, 13.93288372952460, 14.57054136092293
