| L(s) = 1 | + 3·3-s − 2·4-s + 4·7-s + 6·9-s − 6·12-s + 6·19-s + 12·21-s + 9·27-s − 8·28-s − 15·31-s − 12·36-s + 11·37-s − 10·43-s + 9·49-s + 18·57-s + 27·61-s + 24·63-s + 8·64-s + 16·67-s − 3·73-s − 12·76-s − 17·79-s + 9·81-s − 24·84-s − 45·93-s + 27·103-s − 18·108-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 1.51·7-s + 2·9-s − 1.73·12-s + 1.37·19-s + 2.61·21-s + 1.73·27-s − 1.51·28-s − 2.69·31-s − 2·36-s + 1.80·37-s − 1.52·43-s + 9/7·49-s + 2.38·57-s + 3.45·61-s + 3.02·63-s + 64-s + 1.95·67-s − 0.351·73-s − 1.37·76-s − 1.91·79-s + 81-s − 2.61·84-s − 4.66·93-s + 2.66·103-s − 1.73·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.287534141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.287534141\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.20898337722182738029236542287, −10.49831278620155977143497500687, −9.859324735411588998911630704885, −9.704529820792939241649061447679, −9.235219852338695996066870610755, −8.790112898646503023663470037703, −8.426284183633319152442091242206, −8.142644917649185827675823322083, −7.63114268332235906291337470677, −7.23859822324037744774027550113, −6.86686745065389677772433983252, −5.79601223912743460586308788474, −5.24684148744016270174242739152, −4.92459321086285908392015516086, −4.28750309905247473683710449697, −3.76773103021466577869378495194, −3.41195056851713465102309223673, −2.44495539396809546843751774069, −1.95392133223678978025031114901, −1.10319902927824971887898493440,
1.10319902927824971887898493440, 1.95392133223678978025031114901, 2.44495539396809546843751774069, 3.41195056851713465102309223673, 3.76773103021466577869378495194, 4.28750309905247473683710449697, 4.92459321086285908392015516086, 5.24684148744016270174242739152, 5.79601223912743460586308788474, 6.86686745065389677772433983252, 7.23859822324037744774027550113, 7.63114268332235906291337470677, 8.142644917649185827675823322083, 8.426284183633319152442091242206, 8.790112898646503023663470037703, 9.235219852338695996066870610755, 9.704529820792939241649061447679, 9.859324735411588998911630704885, 10.49831278620155977143497500687, 11.20898337722182738029236542287
