| L(s) = 1 | − 2-s + 4-s − 8-s − 2·9-s + 16-s + 2·18-s + 6·25-s − 12·29-s − 32-s − 2·36-s − 4·37-s − 7·49-s − 6·50-s − 4·53-s + 12·58-s + 64-s + 2·72-s + 4·74-s − 5·81-s + 7·98-s + 6·100-s + 4·106-s + 20·109-s − 20·113-s − 12·116-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 1/4·16-s + 0.471·18-s + 6/5·25-s − 2.22·29-s − 0.176·32-s − 1/3·36-s − 0.657·37-s − 49-s − 0.848·50-s − 0.549·53-s + 1.57·58-s + 1/8·64-s + 0.235·72-s + 0.464·74-s − 5/9·81-s + 0.707·98-s + 3/5·100-s + 0.388·106-s + 1.91·109-s − 1.88·113-s − 1.11·116-s + 1/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.858772578072611359526213611131, −8.539034982107098033938033640909, −7.975223574169517042352563461110, −7.46659077481011948645814284815, −7.13989534434007173512612746405, −6.41310543135258797598918521141, −6.13289442032409492126060993659, −5.31164425237226041733774684037, −5.12514243391074363874048515633, −4.19111221198881396430954655287, −3.51782104512240046203049689516, −2.96495953167035953396367353377, −2.19626538646593776391819028602, −1.37922759436015962364017945930, 0,
1.37922759436015962364017945930, 2.19626538646593776391819028602, 2.96495953167035953396367353377, 3.51782104512240046203049689516, 4.19111221198881396430954655287, 5.12514243391074363874048515633, 5.31164425237226041733774684037, 6.13289442032409492126060993659, 6.41310543135258797598918521141, 7.13989534434007173512612746405, 7.46659077481011948645814284815, 7.975223574169517042352563461110, 8.539034982107098033938033640909, 8.858772578072611359526213611131
