L(s) = 1 | + 4·5-s − 3·9-s + 2·25-s + 20·29-s − 12·45-s − 14·49-s − 28·53-s − 12·73-s + 9·81-s + 36·97-s + 4·101-s − 22·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 80·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 9-s + 2/5·25-s + 3.71·29-s − 1.78·45-s − 2·49-s − 3.84·53-s − 1.40·73-s + 81-s + 3.65·97-s + 0.398·101-s − 2·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.64·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.403391428\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.403391428\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−10.79424580459691548950831956254, −10.23421229877843747435058592930, −9.988093934240586032989036843847, −9.317356556158666156286958675354, −8.944850116469648867972974935175, −8.155593376401658883389001410970, −7.86574055998535746161422058704, −6.62912355829218159331214255175, −6.27282080646876620831649653268, −5.98788019309426758968327152208, −5.01452297596172644651029655270, −4.69735185642958092576120623346, −3.23795791772265289886094041940, −2.63895104553179739845027310173, −1.61821392697617915675557729976,
1.61821392697617915675557729976, 2.63895104553179739845027310173, 3.23795791772265289886094041940, 4.69735185642958092576120623346, 5.01452297596172644651029655270, 5.98788019309426758968327152208, 6.27282080646876620831649653268, 6.62912355829218159331214255175, 7.86574055998535746161422058704, 8.155593376401658883389001410970, 8.944850116469648867972974935175, 9.317356556158666156286958675354, 9.988093934240586032989036843847, 10.23421229877843747435058592930, 10.79424580459691548950831956254