| L(s) = 1 | − 3-s − 2·5-s + 9-s − 6·13-s + 2·15-s − 17-s − 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s − 10·37-s + 6·39-s + 6·41-s − 12·43-s − 2·45-s + 51-s + 10·53-s − 8·59-s + 6·61-s + 12·65-s − 12·67-s + 8·69-s + 6·73-s + 75-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 0.242·17-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.960·39-s + 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.140·51-s + 1.37·53-s − 1.04·59-s + 0.768·61-s + 1.48·65-s − 1.46·67-s + 0.963·69-s + 0.702·73-s + 0.115·75-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.64268522619822, −13.40994954191221, −12.43612555847567, −12.23307315591990, −11.84028565970362, −11.71851086440416, −10.89111878703008, −10.29285495867940, −10.12070362118978, −9.604762229717275, −8.928823833598951, −8.278658196956867, −7.938692868881994, −7.514461842013569, −6.782619755334819, −6.636241824957258, −5.865489147588900, −5.265180034669212, −4.806223760708030, −4.287887835215750, −3.890186507930304, −3.108226669908537, −2.498679547737763, −1.907111429567030, −1.053833915001922, 0, 0,
1.053833915001922, 1.907111429567030, 2.498679547737763, 3.108226669908537, 3.890186507930304, 4.287887835215750, 4.806223760708030, 5.265180034669212, 5.865489147588900, 6.636241824957258, 6.782619755334819, 7.514461842013569, 7.938692868881994, 8.278658196956867, 8.928823833598951, 9.604762229717275, 10.12070362118978, 10.29285495867940, 10.89111878703008, 11.71851086440416, 11.84028565970362, 12.23307315591990, 12.43612555847567, 13.40994954191221, 13.64268522619822
