| L(s) = 1 | + 3-s − 7-s − 2·9-s + 6·11-s − 3·13-s − 21-s − 23-s − 5·27-s + 3·29-s − 7·31-s + 6·33-s + 8·37-s − 3·39-s − 11·41-s + 4·43-s − 47-s + 49-s + 4·53-s − 12·59-s + 6·61-s + 2·63-s + 12·67-s − 69-s − 5·71-s − 15·73-s − 6·77-s − 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.80·11-s − 0.832·13-s − 0.218·21-s − 0.208·23-s − 0.962·27-s + 0.557·29-s − 1.25·31-s + 1.04·33-s + 1.31·37-s − 0.480·39-s − 1.71·41-s + 0.609·43-s − 0.145·47-s + 1/7·49-s + 0.549·53-s − 1.56·59-s + 0.768·61-s + 0.251·63-s + 1.46·67-s − 0.120·69-s − 0.593·71-s − 1.75·73-s − 0.683·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.08975706741017, −12.57499776732660, −12.06677875548201, −11.71926900177748, −11.33282682654222, −10.82276531779186, −10.06814029743891, −9.756242846560018, −9.311430109110225, −8.861934527173615, −8.547621177789151, −7.971301323551768, −7.302444747570115, −7.044359332048341, −6.415699339422923, −5.952804436022780, −5.545432120281707, −4.734845984174375, −4.328323393007254, −3.683708728470948, −3.314796988083614, −2.746930562649400, −2.095818363318220, −1.589671125700213, −0.7876115432336750, 0,
0.7876115432336750, 1.589671125700213, 2.095818363318220, 2.746930562649400, 3.314796988083614, 3.683708728470948, 4.328323393007254, 4.734845984174375, 5.545432120281707, 5.952804436022780, 6.415699339422923, 7.044359332048341, 7.302444747570115, 7.971301323551768, 8.547621177789151, 8.861934527173615, 9.311430109110225, 9.756242846560018, 10.06814029743891, 10.82276531779186, 11.33282682654222, 11.71926900177748, 12.06677875548201, 12.57499776732660, 13.08975706741017
