| L(s) = 1 | + 8·11-s + 8·19-s − 4·29-s − 16·31-s + 12·41-s + 14·49-s + 24·59-s + 28·61-s − 16·71-s + 16·79-s + 20·89-s − 12·101-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 1.83·19-s − 0.742·29-s − 2.87·31-s + 1.87·41-s + 2·49-s + 3.12·59-s + 3.58·61-s − 1.89·71-s + 1.80·79-s + 2.11·89-s − 1.19·101-s + 3.44·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.438113539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.438113539\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.402802984915132771648375843219, −9.094743344919859877060610482073, −8.796132632499125503946783245918, −8.528587581033797806845045095530, −7.59962791594882709441493335944, −7.56649741167929220289857634643, −6.98845760616785178267883674835, −6.93918352503521950128550326838, −6.26622151735023857247601604223, −5.77910362268210600002936997576, −5.44699920865496374295878712759, −5.20999346687128674767317548152, −4.33697446811332180746825328684, −3.96417219941928330908910833748, −3.52669041071811231590873734600, −3.50104030966001246223195561515, −2.28470657279009699079811902363, −2.12072137395380159076555044239, −1.12188486750357221554100410567, −0.858435137519986371005479529244,
0.858435137519986371005479529244, 1.12188486750357221554100410567, 2.12072137395380159076555044239, 2.28470657279009699079811902363, 3.50104030966001246223195561515, 3.52669041071811231590873734600, 3.96417219941928330908910833748, 4.33697446811332180746825328684, 5.20999346687128674767317548152, 5.44699920865496374295878712759, 5.77910362268210600002936997576, 6.26622151735023857247601604223, 6.93918352503521950128550326838, 6.98845760616785178267883674835, 7.56649741167929220289857634643, 7.59962791594882709441493335944, 8.528587581033797806845045095530, 8.796132632499125503946783245918, 9.094743344919859877060610482073, 9.402802984915132771648375843219
