| L(s) = 1 | − 2·2-s + 3·4-s + 4·5-s − 4·8-s − 9-s − 8·10-s + 5·16-s + 2·18-s + 12·20-s + 2·25-s + 4·31-s − 6·32-s − 3·36-s + 12·37-s − 16·40-s − 8·41-s + 8·43-s − 4·45-s − 49-s − 4·50-s − 12·59-s − 16·61-s − 8·62-s + 7·64-s + 4·72-s + 20·73-s − 24·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.41·8-s − 1/3·9-s − 2.52·10-s + 5/4·16-s + 0.471·18-s + 2.68·20-s + 2/5·25-s + 0.718·31-s − 1.06·32-s − 1/2·36-s + 1.97·37-s − 2.52·40-s − 1.24·41-s + 1.21·43-s − 0.596·45-s − 1/7·49-s − 0.565·50-s − 1.56·59-s − 2.04·61-s − 1.01·62-s + 7/8·64-s + 0.471·72-s + 2.34·73-s − 2.78·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2965284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2965284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.537084925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537084925\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 41 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.608451289161798004818796404023, −9.264229158285421089095512436177, −8.662281026403992661469811046765, −8.644267008605309030013551729096, −7.81334148258004356158541186665, −7.80971384126690279828508104523, −7.25540427100577117569940722156, −6.71340955734571335601981998644, −6.17071680719276140727220091349, −6.12126332920449288911997929306, −5.76133762770865021089612281801, −5.23546132688697419879236468340, −4.61269392386010833313536331308, −4.17109034182542498846162728644, −3.11086484128558782310879360728, −3.06108901736996037579990096682, −2.12002710720056025871149247640, −2.06626781132466217519800232724, −1.35275979495854795980036423769, −0.60301879878664495448848779196,
0.60301879878664495448848779196, 1.35275979495854795980036423769, 2.06626781132466217519800232724, 2.12002710720056025871149247640, 3.06108901736996037579990096682, 3.11086484128558782310879360728, 4.17109034182542498846162728644, 4.61269392386010833313536331308, 5.23546132688697419879236468340, 5.76133762770865021089612281801, 6.12126332920449288911997929306, 6.17071680719276140727220091349, 6.71340955734571335601981998644, 7.25540427100577117569940722156, 7.80971384126690279828508104523, 7.81334148258004356158541186665, 8.644267008605309030013551729096, 8.662281026403992661469811046765, 9.264229158285421089095512436177, 9.608451289161798004818796404023
