| L(s) = 1 | + 7·3-s + 5-s + 22·9-s + 11·11-s − 12·13-s + 7·15-s − 2·17-s − 82·19-s + 7·23-s − 124·25-s − 35·27-s − 102·29-s + 171·31-s + 77·33-s − 357·37-s − 84·39-s + 114·41-s − 344·43-s + 22·45-s − 96·47-s − 14·51-s − 430·53-s + 11·55-s − 574·57-s + 201·59-s + 2·61-s − 12·65-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.0894·5-s + 0.814·9-s + 0.301·11-s − 0.256·13-s + 0.120·15-s − 0.0285·17-s − 0.990·19-s + 0.0634·23-s − 0.991·25-s − 0.249·27-s − 0.653·29-s + 0.990·31-s + 0.406·33-s − 1.58·37-s − 0.344·39-s + 0.434·41-s − 1.21·43-s + 0.0728·45-s − 0.297·47-s − 0.0384·51-s − 1.11·53-s + 0.0269·55-s − 1.33·57-s + 0.443·59-s + 0.00419·61-s − 0.0228·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 82 T + p^{3} T^{2} \) |
| 23 | \( 1 - 7 T + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 - 171 T + p^{3} T^{2} \) |
| 37 | \( 1 + 357 T + p^{3} T^{2} \) |
| 41 | \( 1 - 114 T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 430 T + p^{3} T^{2} \) |
| 59 | \( 1 - 201 T + p^{3} T^{2} \) |
| 61 | \( 1 - 2 T + p^{3} T^{2} \) |
| 67 | \( 1 - 313 T + p^{3} T^{2} \) |
| 71 | \( 1 + 579 T + p^{3} T^{2} \) |
| 73 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 494 T + p^{3} T^{2} \) |
| 83 | \( 1 + 748 T + p^{3} T^{2} \) |
| 89 | \( 1 + 457 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1037 T + p^{3} T^{2} \) |
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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
−8.387322673613687110264131767545, −7.77440388785601454469937349856, −6.91172241680443746363745576946, −6.09081363987567309146817714538, −4.99178958205084869486885656009, −4.01640190331717340451455582142, −3.32264247744934936610371393135, −2.34951763303194471905740884651, −1.61992213615516054904280662135, 0,
1.61992213615516054904280662135, 2.34951763303194471905740884651, 3.32264247744934936610371393135, 4.01640190331717340451455582142, 4.99178958205084869486885656009, 6.09081363987567309146817714538, 6.91172241680443746363745576946, 7.77440388785601454469937349856, 8.387322673613687110264131767545
