| L(s) = 1 | − 4·11-s − 4·16-s + 10·29-s + 4·31-s − 24·41-s + 10·49-s − 26·61-s − 24·71-s − 30·79-s − 20·89-s + 6·101-s − 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 16·176-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 16-s + 1.85·29-s + 0.718·31-s − 3.74·41-s + 10/7·49-s − 3.32·61-s − 2.84·71-s − 3.37·79-s − 2.11·89-s + 0.597·101-s − 1.91·109-s − 0.909·121-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 − 0.0769·169-s + 0.0760·173-s + 1.20·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3369795836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3369795836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.926425911074091246199654963462, −8.728326883587922561144396079443, −8.208169284579846589864487189466, −7.72669730564748187918750243858, −7.59855581074883485072932842942, −6.98008745326426751706722393994, −6.68493767190287971990430852255, −6.41535151576267958374406220776, −5.86523153108150287672403174493, −5.46627244284282888526708992679, −4.97142366432850910688848350470, −4.78919398410201311247615146256, −4.23602274296385177622895630489, −3.96580596026260140632211124353, −2.96315788195802517891579086132, −2.91005205076429251752636537055, −2.62866116951003123851670006487, −1.52706109683930013424901328980, −1.51817062995960694710678347531, −0.17806132502989746303093302123,
0.17806132502989746303093302123, 1.51817062995960694710678347531, 1.52706109683930013424901328980, 2.62866116951003123851670006487, 2.91005205076429251752636537055, 2.96315788195802517891579086132, 3.96580596026260140632211124353, 4.23602274296385177622895630489, 4.78919398410201311247615146256, 4.97142366432850910688848350470, 5.46627244284282888526708992679, 5.86523153108150287672403174493, 6.41535151576267958374406220776, 6.68493767190287971990430852255, 6.98008745326426751706722393994, 7.59855581074883485072932842942, 7.72669730564748187918750243858, 8.208169284579846589864487189466, 8.728326883587922561144396079443, 8.926425911074091246199654963462
