| L(s) = 1 | − 3·2-s + 5·4-s − 5·8-s + 9-s + 11-s − 4·13-s + 16-s − 3·18-s − 3·22-s − 2·23-s − 4·25-s + 12·26-s + 2·29-s + 7·32-s + 5·36-s − 18·41-s + 5·44-s + 6·46-s + 16·47-s + 9·49-s + 12·50-s − 20·52-s − 6·58-s − 15·64-s − 5·72-s + 6·79-s − 8·81-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 5/2·4-s − 1.76·8-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1/4·16-s − 0.707·18-s − 0.639·22-s − 0.417·23-s − 4/5·25-s + 2.35·26-s + 0.371·29-s + 1.23·32-s + 5/6·36-s − 2.81·41-s + 0.753·44-s + 0.884·46-s + 2.33·47-s + 9/7·49-s + 1.69·50-s − 2.77·52-s − 0.787·58-s − 1.87·64-s − 0.589·72-s + 0.675·79-s − 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 171 T^{2} + p^{2} T^{4} \) |
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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
−8.488165286437270730401594839756, −7.989949773300283141362357849796, −7.58160274808814998787496019415, −7.17302109493562102412958445666, −6.86111209817108375273552376011, −6.36046765793682467256567685736, −5.65074929653116029605763212523, −5.17115839866373868806154396628, −4.43120944678665957746113263556, −3.96898761787115612800662909995, −3.11390863068111940498613941325, −2.30292363832701496322963685469, −1.89462716154926679076231528824, −1.01380105590921296122083378756, 0,
1.01380105590921296122083378756, 1.89462716154926679076231528824, 2.30292363832701496322963685469, 3.11390863068111940498613941325, 3.96898761787115612800662909995, 4.43120944678665957746113263556, 5.17115839866373868806154396628, 5.65074929653116029605763212523, 6.36046765793682467256567685736, 6.86111209817108375273552376011, 7.17302109493562102412958445666, 7.58160274808814998787496019415, 7.989949773300283141362357849796, 8.488165286437270730401594839756
