| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s + 9-s − 4·10-s − 4·13-s − 16-s − 4·17-s + 18-s + 4·20-s + 2·25-s − 4·26-s − 12·29-s + 5·32-s − 4·34-s − 36-s + 12·37-s + 12·40-s − 4·41-s − 4·45-s + 2·49-s + 2·50-s + 4·52-s + 12·53-s − 12·58-s + 12·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s + 1/3·9-s − 1.26·10-s − 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.894·20-s + 2/5·25-s − 0.784·26-s − 2.22·29-s + 0.883·32-s − 0.685·34-s − 1/6·36-s + 1.97·37-s + 1.89·40-s − 0.624·41-s − 0.596·45-s + 2/7·49-s + 0.282·50-s + 0.554·52-s + 1.64·53-s − 1.57·58-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.06987917382067337680927181024, −10.05246880525653227640097293961, −9.723139098205724899141727122835, −8.975261543279013968212899152873, −8.548353092044889979587894449296, −7.75720782023225106841673661541, −7.46995816184461358823952634116, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −5.31622482563829752699303340190, −4.35461089374668409702872860477, −4.20407903283329927801590362766, −3.53074178558790394542885743604, −2.45250023112523203194083612695, 0,
2.45250023112523203194083612695, 3.53074178558790394542885743604, 4.20407903283329927801590362766, 4.35461089374668409702872860477, 5.31622482563829752699303340190, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 7.46995816184461358823952634116, 7.75720782023225106841673661541, 8.548353092044889979587894449296, 8.975261543279013968212899152873, 9.723139098205724899141727122835, 10.05246880525653227640097293961, 11.06987917382067337680927181024
