| L(s) = 1 | + 3·3-s + 6·9-s − 7·13-s + 8·19-s − 5·25-s + 9·27-s − 2·37-s − 21·39-s + 21·43-s − 13·49-s + 24·57-s + 13·61-s + 27·67-s − 7·73-s − 15·75-s + 21·79-s + 9·81-s + 14·97-s + 2·109-s − 6·111-s − 42·117-s − 22·121-s + 127-s + 63·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s − 1.94·13-s + 1.83·19-s − 25-s + 1.73·27-s − 0.328·37-s − 3.36·39-s + 3.20·43-s − 1.85·49-s + 3.17·57-s + 1.66·61-s + 3.29·67-s − 0.819·73-s − 1.73·75-s + 2.36·79-s + 81-s + 1.42·97-s + 0.191·109-s − 0.569·111-s − 3.88·117-s − 2·121-s + 0.0887·127-s + 5.54·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.828595833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.828595833\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.15453012745025151960118533585, −9.659536967431119004597951152558, −9.364916684387878776398149276319, −9.317100694591655554419584409719, −8.683361438763619641295713803479, −7.967559584936506058514810992965, −7.75322911993066621704666828153, −7.70609156837953006471658643707, −6.90756721653461851753727814564, −6.86203230778466158187211205276, −5.95333614520708761182992257464, −5.33390489848676609159591474132, −5.01257368310793641225620357483, −4.44874045262533097722414051148, −3.69044282197258406462112762381, −3.59384461932084520838730647313, −2.65857943238568447246354432715, −2.49471346951424311112651177277, −1.87432532852981564956394832084, −0.853810798041231389069582736605,
0.853810798041231389069582736605, 1.87432532852981564956394832084, 2.49471346951424311112651177277, 2.65857943238568447246354432715, 3.59384461932084520838730647313, 3.69044282197258406462112762381, 4.44874045262533097722414051148, 5.01257368310793641225620357483, 5.33390489848676609159591474132, 5.95333614520708761182992257464, 6.86203230778466158187211205276, 6.90756721653461851753727814564, 7.70609156837953006471658643707, 7.75322911993066621704666828153, 7.967559584936506058514810992965, 8.683361438763619641295713803479, 9.317100694591655554419584409719, 9.364916684387878776398149276319, 9.659536967431119004597951152558, 10.15453012745025151960118533585
