L(s) = 1 | + 6·9-s − 8·11-s + 2·19-s + 4·29-s + 4·41-s + 14·49-s + 8·59-s − 4·61-s + 16·71-s + 16·79-s + 27·81-s + 28·89-s − 48·99-s − 20·101-s + 20·109-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 + 12·171-s + ⋯ |
L(s) = 1 | + 2·9-s − 2.41·11-s + 0.458·19-s + 0.742·29-s + 0.624·41-s + 2·49-s + 1.04·59-s − 0.512·61-s + 1.89·71-s + 1.80·79-s + 3·81-s + 2.96·89-s − 4.82·99-s − 1.99·101-s + 1.91·109-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 + 0.917·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.240053747\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.240053747\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.508553586204081158508369390371, −8.282863763175222891834331344165, −7.83020251693616215678729509286, −7.54903438877418717676603420192, −7.32378912793155189945924058663, −7.01591283215782530277463636810, −6.42966035251266168333488596484, −6.21209353945122681868230499151, −5.54375321471243224689918990874, −5.25405959809490831572889143135, −4.89411199117978251148315187504, −4.66331988457416655776782236872, −3.98656084744732598759965771930, −3.83164515058154759077597532364, −3.13540596757981450058861795088, −2.72819593184818726856591699654, −2.04618870146160148589825248941, −2.04159280703731431639476203844, −0.922003547097038685831122276390, −0.66631612904177261118759649419,
0.66631612904177261118759649419, 0.922003547097038685831122276390, 2.04159280703731431639476203844, 2.04618870146160148589825248941, 2.72819593184818726856591699654, 3.13540596757981450058861795088, 3.83164515058154759077597532364, 3.98656084744732598759965771930, 4.66331988457416655776782236872, 4.89411199117978251148315187504, 5.25405959809490831572889143135, 5.54375321471243224689918990874, 6.21209353945122681868230499151, 6.42966035251266168333488596484, 7.01591283215782530277463636810, 7.32378912793155189945924058663, 7.54903438877418717676603420192, 7.83020251693616215678729509286, 8.282863763175222891834331344165, 8.508553586204081158508369390371