| L(s) = 1 | + 2·2-s − 5·4-s − 7·7-s − 20·8-s − 34·11-s − 14·14-s + 5·16-s − 68·22-s − 64·23-s + 47·25-s + 35·28-s − 4·29-s + 118·32-s + 16·37-s + 52·43-s + 170·44-s − 128·46-s + 94·50-s + 2·53-s + 140·56-s − 8·58-s + 111·64-s − 68·67-s + 20·71-s + 32·74-s + 238·77-s − 92·79-s + ⋯ |
| L(s) = 1 | + 2-s − 5/4·4-s − 7-s − 5/2·8-s − 3.09·11-s − 14-s + 5/16·16-s − 3.09·22-s − 2.78·23-s + 1.87·25-s + 5/4·28-s − 0.137·29-s + 3.68·32-s + 0.432·37-s + 1.20·43-s + 3.86·44-s − 2.78·46-s + 1.87·50-s + 2/53·53-s + 5/2·56-s − 0.137·58-s + 1.73·64-s − 1.01·67-s + 0.281·71-s + 0.432·74-s + 3.09·77-s − 1.16·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3314551837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3314551837\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 47 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 38 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 394 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1415 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2966 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 7391 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3335 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 13142 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10391 T^{2} + p^{4} T^{4} \) |
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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
−12.77116681215677246141968053360, −12.23117804345371914011053653621, −12.17280929211006599713793801629, −10.99806804712751197822960994793, −10.51929988321912560901238440231, −10.02381456096995340894473142818, −9.785918865459859663624757060411, −9.095117619245174764350320389155, −8.541791010621970061454673428167, −7.914037290470480177414403438918, −7.78231874251645278760711753648, −6.64970874101008954596164589703, −5.93871724284969019484933223568, −5.54032767006235777613325884001, −5.11864176104495110377628319430, −4.42068306404786360271414731977, −3.88813747252085528635158596070, −2.91287068670085263631339624691, −2.68260966030812531205220731248, −0.27855393976472914367043885314,
0.27855393976472914367043885314, 2.68260966030812531205220731248, 2.91287068670085263631339624691, 3.88813747252085528635158596070, 4.42068306404786360271414731977, 5.11864176104495110377628319430, 5.54032767006235777613325884001, 5.93871724284969019484933223568, 6.64970874101008954596164589703, 7.78231874251645278760711753648, 7.914037290470480177414403438918, 8.541791010621970061454673428167, 9.095117619245174764350320389155, 9.785918865459859663624757060411, 10.02381456096995340894473142818, 10.51929988321912560901238440231, 10.99806804712751197822960994793, 12.17280929211006599713793801629, 12.23117804345371914011053653621, 12.77116681215677246141968053360
