| L(s) = 1 | − 4-s − 2·5-s + 9-s − 3·16-s + 2·20-s − 25-s − 36-s − 2·45-s − 6·49-s + 7·64-s − 16·71-s + 6·80-s + 81-s + 28·89-s + 100-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s + 1/3·9-s − 3/4·16-s + 0.447·20-s − 1/5·25-s − 1/6·36-s − 0.298·45-s − 6/7·49-s + 7/8·64-s − 1.89·71-s + 0.670·80-s + 1/9·81-s + 2.96·89-s + 1/10·100-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.37502840345174702922276298069, −6.97793516144482552722802579390, −6.45827470460787329312486520040, −6.18522656249781606647050153553, −5.54486807324915564164405031200, −5.13498488652535083964987208368, −4.59835288593959400153873761477, −4.36454563276548543570477802654, −3.91214177750804457483802607233, −3.39305197403574913212767194111, −2.97484146721407084361984934359, −2.21344623561431793817934308120, −1.69513758921887876847281638863, −0.78150154220866768360452665462, 0,
0.78150154220866768360452665462, 1.69513758921887876847281638863, 2.21344623561431793817934308120, 2.97484146721407084361984934359, 3.39305197403574913212767194111, 3.91214177750804457483802607233, 4.36454563276548543570477802654, 4.59835288593959400153873761477, 5.13498488652535083964987208368, 5.54486807324915564164405031200, 6.18522656249781606647050153553, 6.45827470460787329312486520040, 6.97793516144482552722802579390, 7.37502840345174702922276298069
