| L(s) = 1 | + 3-s + 3·7-s + 3·21-s − 2·25-s − 27-s + 43-s + 5·49-s − 61-s − 3·67-s − 2·75-s + 2·79-s − 81-s + 3·97-s − 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 5·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + ⋯ |
| L(s) = 1 | + 3-s + 3·7-s + 3·21-s − 2·25-s − 27-s + 43-s + 5·49-s − 61-s − 3·67-s − 2·75-s + 2·79-s − 81-s + 3·97-s − 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 5·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.409793294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.409793294\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.284804723633816577482141140160, −8.919463747275911891923516760064, −8.844173110394472781463167415039, −8.230181683718558244925416899976, −7.948950295788497112579356918428, −7.65878522778450404790269942355, −7.60706026699309171868249270364, −7.07046159732048141288919066605, −6.13655952796112925254041787196, −6.03454453587760628266545405999, −5.39871823781162242755620560095, −5.10402976246783382971351336778, −4.59673344106492540293986117982, −4.24421059907432348318115350246, −3.87865735450143787661130835212, −3.24547977871478479452233466290, −2.60803587655338948401594424580, −1.99761782792138409732538620945, −1.85710015148427093512776057106, −1.19736238697406130626784494067,
1.19736238697406130626784494067, 1.85710015148427093512776057106, 1.99761782792138409732538620945, 2.60803587655338948401594424580, 3.24547977871478479452233466290, 3.87865735450143787661130835212, 4.24421059907432348318115350246, 4.59673344106492540293986117982, 5.10402976246783382971351336778, 5.39871823781162242755620560095, 6.03454453587760628266545405999, 6.13655952796112925254041787196, 7.07046159732048141288919066605, 7.60706026699309171868249270364, 7.65878522778450404790269942355, 7.948950295788497112579356918428, 8.230181683718558244925416899976, 8.844173110394472781463167415039, 8.919463747275911891923516760064, 9.284804723633816577482141140160
