| L(s) = 1 | + 2-s + 4-s − 4·5-s + 8-s + 9-s − 4·10-s − 4·13-s + 16-s + 2·17-s + 18-s − 4·20-s + 2·25-s − 4·26-s − 20·29-s + 32-s + 2·34-s + 36-s − 4·37-s − 4·40-s + 20·41-s − 4·45-s − 14·49-s + 2·50-s − 4·52-s + 12·53-s − 20·58-s − 20·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.894·20-s + 2/5·25-s − 0.784·26-s − 3.71·29-s + 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.657·37-s − 0.632·40-s + 3.12·41-s − 0.596·45-s − 2·49-s + 0.282·50-s − 0.554·52-s + 1.64·53-s − 2.62·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.570394810938145814587700544156, −9.080746654144001493159861980940, −7.972624440390951709752181868622, −7.958882066985845549410362974675, −7.36440720073199929079936076774, −7.22142790948264379459326388401, −6.37869009161194820234692371426, −5.51457882926074302977186886250, −5.36244298721167160718725340278, −4.24541437163957543556604196065, −4.13875090005906653574798019221, −3.56656081546400251035965478912, −2.76309813409348720224135008444, −1.78155952274398559703453953653, 0,
1.78155952274398559703453953653, 2.76309813409348720224135008444, 3.56656081546400251035965478912, 4.13875090005906653574798019221, 4.24541437163957543556604196065, 5.36244298721167160718725340278, 5.51457882926074302977186886250, 6.37869009161194820234692371426, 7.22142790948264379459326388401, 7.36440720073199929079936076774, 7.958882066985845549410362974675, 7.972624440390951709752181868622, 9.080746654144001493159861980940, 9.570394810938145814587700544156
