L(s) = 1 | + 3·3-s − 2·5-s + 35·7-s + 18·11-s + 66·13-s − 6·15-s + 68·17-s − 25·19-s + 105·21-s − 92·23-s + 125·25-s − 27·27-s + 184·29-s − 25·31-s + 54·33-s − 70·35-s + 213·37-s + 198·39-s + 188·41-s − 134·43-s − 278·47-s + 882·49-s + 204·51-s + 400·53-s − 36·55-s − 75·57-s − 744·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.178·5-s + 1.88·7-s + 0.493·11-s + 1.40·13-s − 0.103·15-s + 0.970·17-s − 0.301·19-s + 1.09·21-s − 0.834·23-s + 25-s − 0.192·27-s + 1.17·29-s − 0.144·31-s + 0.284·33-s − 0.338·35-s + 0.946·37-s + 0.812·39-s + 0.716·41-s − 0.475·43-s − 0.862·47-s + 18/7·49-s + 0.560·51-s + 1.03·53-s − 0.0882·55-s − 0.174·57-s − 1.64·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.177718767\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.177718767\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - 121 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T - 1007 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 33 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 p T - p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T - 6234 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 p T - 7 p^{2} T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 25 T - 29166 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 213 T - 5284 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 94 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 67 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 278 T - 26539 T^{2} + 278 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 400 T + 11123 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 744 T + 348157 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 734 T + 311775 T^{2} - 734 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 555 T + 7262 T^{2} + 555 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 642 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 973 T + 557712 T^{2} + 973 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 785 T + 123186 T^{2} - 785 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 822 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 424 T - 525193 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 734 T + p^{3} T^{2} )^{2} \) |
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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
−12.43640495540356873268671480603, −12.06333618662214552821539035762, −11.43758551122152971292876806008, −11.27052678409937702414855955124, −10.58183272967781755286407685014, −10.22197667793205238802035459152, −9.458119420793879734094666438344, −8.757686137049928052091729091563, −8.340010853066541709275023503331, −8.256394316350968245010379053760, −7.47500484261877181607105383178, −6.98140600752645564681399916220, −5.95390611052833781534537451694, −5.74669477323666047790154147731, −4.61615489084740071455955144058, −4.39768393450666439888051214931, −3.52227183615706175947489380356, −2.70876883768963688033426113023, −1.61647319045703788701912693309, −1.07902011614553030221086211987,
1.07902011614553030221086211987, 1.61647319045703788701912693309, 2.70876883768963688033426113023, 3.52227183615706175947489380356, 4.39768393450666439888051214931, 4.61615489084740071455955144058, 5.74669477323666047790154147731, 5.95390611052833781534537451694, 6.98140600752645564681399916220, 7.47500484261877181607105383178, 8.256394316350968245010379053760, 8.340010853066541709275023503331, 8.757686137049928052091729091563, 9.458119420793879734094666438344, 10.22197667793205238802035459152, 10.58183272967781755286407685014, 11.27052678409937702414855955124, 11.43758551122152971292876806008, 12.06333618662214552821539035762, 12.43640495540356873268671480603