L(s) = 1 | − 3-s − 4-s − 2·7-s + 12-s − 3·16-s − 6·17-s + 4·19-s + 2·21-s − 6·23-s − 5·25-s + 4·27-s + 2·28-s − 8·37-s + 3·48-s − 2·49-s + 6·51-s − 4·57-s − 12·59-s + 7·64-s + 6·68-s + 6·69-s + 16·73-s + 5·75-s − 4·76-s − 7·81-s − 2·84-s + 6·92-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.755·7-s + 0.288·12-s − 3/4·16-s − 1.45·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s − 25-s + 0.769·27-s + 0.377·28-s − 1.31·37-s + 0.433·48-s − 2/7·49-s + 0.840·51-s − 0.529·57-s − 1.56·59-s + 7/8·64-s + 0.727·68-s + 0.722·69-s + 1.87·73-s + 0.577·75-s − 0.458·76-s − 7/9·81-s − 0.218·84-s + 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.64633384544385515680215023732, −9.869886958627660833094493302004, −9.518419820493640327168381820105, −9.066172426018826794796087207332, −8.372850595683571950500622849438, −7.88521058464608532391749024144, −6.89517010030384008110494764675, −6.71103886476802877027475809592, −5.94515789337932256796297517782, −5.36848475646101974564338342579, −4.59172082351411232785791696242, −4.03324695467829362811994557437, −3.13165478752451171735177074911, −2.02226205952559193812136055481, 0,
2.02226205952559193812136055481, 3.13165478752451171735177074911, 4.03324695467829362811994557437, 4.59172082351411232785791696242, 5.36848475646101974564338342579, 5.94515789337932256796297517782, 6.71103886476802877027475809592, 6.89517010030384008110494764675, 7.88521058464608532391749024144, 8.372850595683571950500622849438, 9.066172426018826794796087207332, 9.518419820493640327168381820105, 9.869886958627660833094493302004, 10.64633384544385515680215023732