| L(s) = 1 | + 2·3-s − 6·7-s + 9-s − 8·13-s − 2·19-s − 12·21-s − 9·25-s − 4·27-s + 8·31-s + 20·37-s − 16·39-s + 2·43-s + 13·49-s − 4·57-s − 26·61-s − 6·63-s − 24·67-s + 18·73-s − 18·75-s + 16·79-s − 11·81-s + 48·91-s + 16·93-s − 16·97-s − 12·103-s + 40·111-s − 8·117-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.26·7-s + 1/3·9-s − 2.21·13-s − 0.458·19-s − 2.61·21-s − 9/5·25-s − 0.769·27-s + 1.43·31-s + 3.28·37-s − 2.56·39-s + 0.304·43-s + 13/7·49-s − 0.529·57-s − 3.32·61-s − 0.755·63-s − 2.93·67-s + 2.10·73-s − 2.07·75-s + 1.80·79-s − 1.22·81-s + 5.03·91-s + 1.65·93-s − 1.62·97-s − 1.18·103-s + 3.79·111-s − 0.739·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−9.559628331745101664415344236205, −9.553328544165326093753901438762, −9.084561409834522310688102275122, −8.146906664648363880391573042940, −7.62429229152375892340957919668, −7.53862789028003683931489821585, −6.44323514092267885981017350010, −6.33560598867710411281522365718, −5.65147980657085516848049895381, −4.55834684131140031321784081525, −4.15537276955970084853193543262, −3.16174551478824360442184908445, −2.82511261070703297705797963037, −2.25610582463945759407900622794, 0,
2.25610582463945759407900622794, 2.82511261070703297705797963037, 3.16174551478824360442184908445, 4.15537276955970084853193543262, 4.55834684131140031321784081525, 5.65147980657085516848049895381, 6.33560598867710411281522365718, 6.44323514092267885981017350010, 7.53862789028003683931489821585, 7.62429229152375892340957919668, 8.146906664648363880391573042940, 9.084561409834522310688102275122, 9.553328544165326093753901438762, 9.559628331745101664415344236205
