| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 4·11-s − 6·13-s − 4·15-s + 4·17-s + 6·19-s + 4·23-s − 25-s − 4·27-s − 6·29-s − 4·31-s − 8·33-s − 6·37-s − 12·39-s − 4·41-s − 12·43-s − 2·45-s − 12·47-s + 8·51-s + 6·53-s + 8·55-s + 12·57-s + 6·59-s + 6·61-s + 12·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.03·15-s + 0.970·17-s + 1.37·19-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.39·33-s − 0.986·37-s − 1.92·39-s − 0.624·41-s − 1.82·43-s − 0.298·45-s − 1.75·47-s + 1.12·51-s + 0.824·53-s + 1.07·55-s + 1.58·57-s + 0.781·59-s + 0.768·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.961369659678962199687138816425, −8.067591524836783647495825405397, −7.56797051521186077953967862619, −7.15163819767907692733871434919, −5.41868105719446805469310308644, −4.94424227501604500687383311520, −3.48865751943528789951142169716, −3.12124281504481352218343737067, −1.99002816982159760744296199407, 0,
1.99002816982159760744296199407, 3.12124281504481352218343737067, 3.48865751943528789951142169716, 4.94424227501604500687383311520, 5.41868105719446805469310308644, 7.15163819767907692733871434919, 7.56797051521186077953967862619, 8.067591524836783647495825405397, 8.961369659678962199687138816425
