| L(s) = 1 | + 2-s + 4-s + 8-s + 4·9-s + 2·13-s + 16-s − 15·17-s + 4·18-s − 5·25-s + 2·26-s + 32-s − 15·34-s + 4·36-s + 2·41-s − 5·49-s − 5·50-s + 2·52-s + 5·53-s − 12·61-s + 64-s − 15·68-s + 4·72-s − 6·73-s + 7·81-s + 2·82-s − 21·89-s + 12·97-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 4/3·9-s + 0.554·13-s + 1/4·16-s − 3.63·17-s + 0.942·18-s − 25-s + 0.392·26-s + 0.176·32-s − 2.57·34-s + 2/3·36-s + 0.312·41-s − 5/7·49-s − 0.707·50-s + 0.277·52-s + 0.686·53-s − 1.53·61-s + 1/8·64-s − 1.81·68-s + 0.471·72-s − 0.702·73-s + 7/9·81-s + 0.220·82-s − 2.22·89-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.85352427209644252529875878279, −7.12326560527005019959983901029, −6.99608647630267093889034553384, −6.50482716048609744203410300214, −6.20509494638406742355830022524, −5.67833703367200168513113101759, −4.98540935785926339874018105138, −4.51185622749849948715272631835, −4.20485736101064834491331654886, −3.97713276546627912326365603184, −3.16050582969469397645827641236, −2.43068626843707114376811070245, −1.98823024842304050204700776123, −1.39584241770973838484214809799, 0,
1.39584241770973838484214809799, 1.98823024842304050204700776123, 2.43068626843707114376811070245, 3.16050582969469397645827641236, 3.97713276546627912326365603184, 4.20485736101064834491331654886, 4.51185622749849948715272631835, 4.98540935785926339874018105138, 5.67833703367200168513113101759, 6.20509494638406742355830022524, 6.50482716048609744203410300214, 6.99608647630267093889034553384, 7.12326560527005019959983901029, 7.85352427209644252529875878279
