| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 9-s + 2·11-s − 3·13-s − 2·14-s + 16-s − 18-s − 2·22-s − 9·25-s + 3·26-s + 2·28-s − 8·31-s − 32-s + 36-s + 4·43-s + 2·44-s + 16·47-s − 3·49-s + 9·50-s − 3·52-s − 2·56-s − 8·61-s + 8·62-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.832·13-s − 0.534·14-s + 1/4·16-s − 0.235·18-s − 0.426·22-s − 9/5·25-s + 0.588·26-s + 0.377·28-s − 1.43·31-s − 0.176·32-s + 1/6·36-s + 0.609·43-s + 0.301·44-s + 2.33·47-s − 3/7·49-s + 1.27·50-s − 0.416·52-s − 0.267·56-s − 1.02·61-s + 1.01·62-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 107 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 155 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
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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
−8.052568045208381359677937374982, −7.57320228937667987122803861691, −7.23181733152527249994055373322, −7.05658982236984914418500835937, −6.17975579228866752883442672135, −5.84511383785497292243149766874, −5.48132910360865898684206051690, −4.73616248677995112988085325932, −4.28852871871565590883410461135, −3.84616135040177657770178497079, −3.14278177612352520034403260705, −2.35662710373285502549905990334, −1.87074288129605519169823770303, −1.23698541781621912625690937688, 0,
1.23698541781621912625690937688, 1.87074288129605519169823770303, 2.35662710373285502549905990334, 3.14278177612352520034403260705, 3.84616135040177657770178497079, 4.28852871871565590883410461135, 4.73616248677995112988085325932, 5.48132910360865898684206051690, 5.84511383785497292243149766874, 6.17975579228866752883442672135, 7.05658982236984914418500835937, 7.23181733152527249994055373322, 7.57320228937667987122803861691, 8.052568045208381359677937374982
