| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s − 2·14-s − 16-s − 10·17-s + 6·23-s + 8·25-s − 2·28-s − 10·31-s − 5·32-s + 10·34-s − 4·41-s − 6·46-s − 6·47-s − 3·49-s − 8·50-s + 6·56-s + 10·62-s + 7·64-s + 10·68-s − 8·71-s − 4·73-s + 6·79-s + 4·82-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 0.534·14-s − 1/4·16-s − 2.42·17-s + 1.25·23-s + 8/5·25-s − 0.377·28-s − 1.79·31-s − 0.883·32-s + 1.71·34-s − 0.624·41-s − 0.884·46-s − 0.875·47-s − 3/7·49-s − 1.13·50-s + 0.801·56-s + 1.27·62-s + 7/8·64-s + 1.21·68-s − 0.949·71-s − 0.468·73-s + 0.675·79-s + 0.441·82-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.871415429730867931626019060088, −8.310836747793656367534705861594, −7.981273041406930648833538630199, −7.22967943932231695135318750774, −6.93225845707495568453488322983, −6.58582320781888110701679481137, −5.71203585560334225029412443552, −5.01361926234580575550834509635, −4.83452918732394601916818109792, −4.31423602400404044304388677178, −3.63333331448173413922530583590, −2.80952555503907413890783276275, −1.98154620057462151554929555738, −1.30433199199338668521553453852, 0,
1.30433199199338668521553453852, 1.98154620057462151554929555738, 2.80952555503907413890783276275, 3.63333331448173413922530583590, 4.31423602400404044304388677178, 4.83452918732394601916818109792, 5.01361926234580575550834509635, 5.71203585560334225029412443552, 6.58582320781888110701679481137, 6.93225845707495568453488322983, 7.22967943932231695135318750774, 7.981273041406930648833538630199, 8.310836747793656367534705861594, 8.871415429730867931626019060088
