| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s + 10·9-s − 4·13-s − 4·16-s − 40·18-s − 8·25-s + 16·26-s + 20·29-s + 32·32-s + 80·36-s − 12·41-s + 32·50-s − 32·52-s − 80·58-s − 64·64-s − 80·72-s − 4·73-s + 57·81-s + 48·82-s − 64·100-s − 32·101-s + 32·104-s + 160·116-s − 40·117-s − 16·121-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 2.82·8-s + 10/3·9-s − 1.10·13-s − 16-s − 9.42·18-s − 8/5·25-s + 3.13·26-s + 3.71·29-s + 5.65·32-s + 40/3·36-s − 1.87·41-s + 4.52·50-s − 4.43·52-s − 10.5·58-s − 8·64-s − 9.42·72-s − 0.468·73-s + 19/3·81-s + 5.30·82-s − 6.39·100-s − 3.18·101-s + 3.13·104-s + 14.8·116-s − 3.69·117-s − 1.45·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2752625016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2752625016\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 180 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.20502480746488792986201362112, −10.09274116297637992688470111299, −9.825539837070753814876225330174, −9.618615021447665228745127484935, −9.367053502642331049008925863655, −9.247705733171638758499153361632, −8.498728099871941280755118620706, −8.383404467759034348062976201851, −8.193847415573616937672891255914, −7.86073020523588095054832532109, −7.43859718267906602219076151765, −7.26909014234119246530747526477, −7.02967586411552443769009147024, −6.62028301150203796317331171312, −6.61271282106728524039149849772, −6.09537738027108013529802472511, −5.08997692610588456024831950803, −5.00498835510093202619703988026, −4.47854774855280332112947046928, −4.26217687523177261315411425500, −3.87646399942521206091126630017, −2.88379729438543074857811845954, −2.23664913918173950709089547972, −1.62806963452952433286508874001, −1.17512742785760637790603110853,
1.17512742785760637790603110853, 1.62806963452952433286508874001, 2.23664913918173950709089547972, 2.88379729438543074857811845954, 3.87646399942521206091126630017, 4.26217687523177261315411425500, 4.47854774855280332112947046928, 5.00498835510093202619703988026, 5.08997692610588456024831950803, 6.09537738027108013529802472511, 6.61271282106728524039149849772, 6.62028301150203796317331171312, 7.02967586411552443769009147024, 7.26909014234119246530747526477, 7.43859718267906602219076151765, 7.86073020523588095054832532109, 8.193847415573616937672891255914, 8.383404467759034348062976201851, 8.498728099871941280755118620706, 9.247705733171638758499153361632, 9.367053502642331049008925863655, 9.618615021447665228745127484935, 9.825539837070753814876225330174, 10.09274116297637992688470111299, 10.20502480746488792986201362112
