| L(s) = 1 | + 8·5-s − 12·13-s + 16·17-s + 38·25-s − 8·29-s − 4·37-s − 16·41-s − 14·49-s − 8·53-s − 20·61-s − 96·65-s + 12·73-s + 128·85-s + 32·89-s − 36·97-s − 40·101-s − 12·109-s + 32·113-s − 22·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 3.32·13-s + 3.88·17-s + 38/5·25-s − 1.48·29-s − 0.657·37-s − 2.49·41-s − 2·49-s − 1.09·53-s − 2.56·61-s − 11.9·65-s + 1.40·73-s + 13.8·85-s + 3.39·89-s − 3.65·97-s − 3.98·101-s − 1.14·109-s + 3.01·113-s − 2·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.646260255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.646260255\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−14.39049550721371311483402596998, −13.68986811781796257702132719939, −13.68986811781796257702132719939, −12.71105456832920786404928308463, −12.71105456832920786404928308463, −12.01980574446063071553934149258, −12.01980574446063071553934149258, −10.55965367152125573686069092164, −10.55965367152125573686069092164, −9.816898356311968022813901953755, −9.816898356311968022813901953755, −9.372732192062624224981148085357, −9.372732192062624224981148085357, −7.909987862286350183375095910259, −7.909987862286350183375095910259, −6.81411763071537323600840753231, −6.81411763071537323600840753231, −5.66990658847027277525585177583, −5.66990658847027277525585177583, −5.01237677697104957198001932438, −5.01237677697104957198001932438, −3.02473421793484999957631922902, −3.02473421793484999957631922902, −1.74502909344688296171382532979, −1.74502909344688296171382532979,
1.74502909344688296171382532979, 1.74502909344688296171382532979, 3.02473421793484999957631922902, 3.02473421793484999957631922902, 5.01237677697104957198001932438, 5.01237677697104957198001932438, 5.66990658847027277525585177583, 5.66990658847027277525585177583, 6.81411763071537323600840753231, 6.81411763071537323600840753231, 7.909987862286350183375095910259, 7.909987862286350183375095910259, 9.372732192062624224981148085357, 9.372732192062624224981148085357, 9.816898356311968022813901953755, 9.816898356311968022813901953755, 10.55965367152125573686069092164, 10.55965367152125573686069092164, 12.01980574446063071553934149258, 12.01980574446063071553934149258, 12.71105456832920786404928308463, 12.71105456832920786404928308463, 13.68986811781796257702132719939, 13.68986811781796257702132719939, 14.39049550721371311483402596998
