| L(s) = 1 | − 6·5-s − 2·13-s − 16·17-s + 18·25-s − 14·29-s − 10·37-s + 14·49-s − 10·53-s − 2·61-s + 12·65-s − 9·81-s + 96·85-s + 16·97-s + 22·101-s − 14·109-s − 28·113-s − 30·125-s + 127-s + 131-s + 137-s + 139-s + 84·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 0.554·13-s − 3.88·17-s + 18/5·25-s − 2.59·29-s − 1.64·37-s + 2·49-s − 1.37·53-s − 0.256·61-s + 1.48·65-s − 81-s + 10.4·85-s + 1.62·97-s + 2.18·101-s − 1.34·109-s − 2.63·113-s − 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.97·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.72756617107388021907404876625, −10.68086206437723389100626391399, −9.694846797595585628577505532914, −9.069540656469781468452912457701, −8.763736629208434220392247529823, −8.632234507736590101961628433772, −7.71844317111024991511995539038, −7.66700851176729134844715704013, −7.03123875703169337427194047700, −6.86681513196180977535606462490, −6.20005292841791693694369192528, −5.35429359836309575592434763750, −4.64151812143918439234263580506, −4.44215588362433810879658579021, −3.78391857123772335592102552475, −3.63749608110412046786385718474, −2.57789197994211068748399544012, −1.94235188945846926758769502317, 0, 0,
1.94235188945846926758769502317, 2.57789197994211068748399544012, 3.63749608110412046786385718474, 3.78391857123772335592102552475, 4.44215588362433810879658579021, 4.64151812143918439234263580506, 5.35429359836309575592434763750, 6.20005292841791693694369192528, 6.86681513196180977535606462490, 7.03123875703169337427194047700, 7.66700851176729134844715704013, 7.71844317111024991511995539038, 8.632234507736590101961628433772, 8.763736629208434220392247529823, 9.069540656469781468452912457701, 9.694846797595585628577505532914, 10.68086206437723389100626391399, 10.72756617107388021907404876625
