L(s) = 1 | − 2·5-s + 8·13-s − 4·17-s + 3·25-s − 20·29-s − 10·49-s + 12·53-s − 28·61-s − 16·65-s + 12·73-s + 8·85-s − 24·89-s − 28·97-s + 12·101-s + 4·109-s + 36·113-s − 18·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 2.21·13-s − 0.970·17-s + 3/5·25-s − 3.71·29-s − 1.42·49-s + 1.64·53-s − 3.58·61-s − 1.98·65-s + 1.40·73-s + 0.867·85-s − 2.54·89-s − 2.84·97-s + 1.19·101-s + 0.383·109-s + 3.38·113-s − 1.63·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.687154242699380983384182548901, −8.273767264975110551561533420364, −7.70282942257969207736023389997, −7.41534942734669416840578685752, −6.75835720857323144112923149074, −6.36049438647887892819733581821, −5.65023019086191961009411991273, −5.52198466275353226488973122049, −4.40097747366580162117975084188, −4.21999895332902477463481579150, −3.48524593863450834073159469636, −3.26490524893388333289715416456, −2.06566091715443158914144420460, −1.39528505582892974646328461150, 0,
1.39528505582892974646328461150, 2.06566091715443158914144420460, 3.26490524893388333289715416456, 3.48524593863450834073159469636, 4.21999895332902477463481579150, 4.40097747366580162117975084188, 5.52198466275353226488973122049, 5.65023019086191961009411991273, 6.36049438647887892819733581821, 6.75835720857323144112923149074, 7.41534942734669416840578685752, 7.70282942257969207736023389997, 8.273767264975110551561533420364, 8.687154242699380983384182548901