| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 2·11-s − 12-s − 4·13-s − 16-s − 18-s + 2·22-s − 16·23-s + 3·24-s − 6·25-s + 4·26-s + 27-s − 5·32-s − 2·33-s − 36-s + 12·37-s − 4·39-s + 2·44-s + 16·46-s − 16·47-s − 48-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s − 0.235·18-s + 0.426·22-s − 3.33·23-s + 0.612·24-s − 6/5·25-s + 0.784·26-s + 0.192·27-s − 0.883·32-s − 0.348·33-s − 1/6·36-s + 1.97·37-s − 0.640·39-s + 0.301·44-s + 2.35·46-s − 2.33·47-s − 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.723139098205724899141727122835, −9.637990407765945223554197990480, −8.548353092044889979587894449296, −8.389268452519734683124359977272, −7.75720782023225106841673661541, −7.60731726518247812897599595849, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −5.64913594526162848677189952595, −4.52395871965258729630923756023, −4.35461089374668409702872860477, −3.53074178558790394542885743604, −2.45250023112523203194083612695, −1.82001999803537272093437549398, 0,
1.82001999803537272093437549398, 2.45250023112523203194083612695, 3.53074178558790394542885743604, 4.35461089374668409702872860477, 4.52395871965258729630923756023, 5.64913594526162848677189952595, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 7.60731726518247812897599595849, 7.75720782023225106841673661541, 8.389268452519734683124359977272, 8.548353092044889979587894449296, 9.637990407765945223554197990480, 9.723139098205724899141727122835
