| L(s) = 1 | − 2·3-s + 5-s + 9-s + 3·11-s − 13-s − 2·15-s + 2·17-s + 19-s − 23-s + 25-s + 4·27-s + 2·29-s − 4·31-s − 6·33-s − 9·37-s + 2·39-s − 3·41-s − 2·43-s + 45-s + 9·47-s − 4·51-s − 9·53-s + 3·55-s − 2·57-s + 12·61-s − 65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 0.229·19-s − 0.208·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s − 1.04·33-s − 1.47·37-s + 0.320·39-s − 0.468·41-s − 0.304·43-s + 0.149·45-s + 1.31·47-s − 0.560·51-s − 1.23·53-s + 0.404·55-s − 0.264·57-s + 1.53·61-s − 0.124·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.19599992941458, −16.06029594571461, −15.17322086136659, −14.46865048459416, −14.16311440762477, −13.51282639035190, −12.73943162384081, −12.23077328275939, −11.83433767549388, −11.28166465116962, −10.66067930801240, −10.13487342931340, −9.583293562498507, −8.865752848543958, −8.345564586047552, −7.352532417224955, −6.859257335051778, −6.285891799250391, −5.606444117400648, −5.246745762238084, −4.463007864750319, −3.683001648006506, −2.865875628650492, −1.812945726301638, −1.067325244287093, 0,
1.067325244287093, 1.812945726301638, 2.865875628650492, 3.683001648006506, 4.463007864750319, 5.246745762238084, 5.606444117400648, 6.285891799250391, 6.859257335051778, 7.352532417224955, 8.345564586047552, 8.865752848543958, 9.583293562498507, 10.13487342931340, 10.66067930801240, 11.28166465116962, 11.83433767549388, 12.23077328275939, 12.73943162384081, 13.51282639035190, 14.16311440762477, 14.46865048459416, 15.17322086136659, 16.06029594571461, 16.19599992941458
