| L(s) = 1 | − 2·3-s + 9-s + 12·11-s + 10·25-s + 4·27-s − 24·33-s + 14·49-s − 12·59-s + 4·73-s − 20·75-s − 11·81-s + 36·83-s + 20·97-s + 12·99-s − 12·107-s + 86·121-s + 127-s + 131-s + 137-s + 139-s − 28·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 3.61·11-s + 2·25-s + 0.769·27-s − 4.17·33-s + 2·49-s − 1.56·59-s + 0.468·73-s − 2.30·75-s − 1.22·81-s + 3.95·83-s + 2.03·97-s + 1.20·99-s − 1.16·107-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.30·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.831641843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.831641843\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.46033345850192554351588473598, −10.41028537391621499623050381498, −9.529679604819364368860346342313, −9.114235929380825439842956604773, −9.049635266734728437059057907380, −8.663013357009031343385744120378, −7.933233395447223951528066045261, −7.29678260320644648977236916626, −6.73788536351109334688842912040, −6.68738746488112880918032304464, −6.09965473708627903733411870985, −5.97665725329863433223916161740, −4.97038504677694939157559048920, −4.85296615641386915626758451398, −4.10097693297599374182001353842, −3.75030586513910626996503822543, −3.17748289583813502672208186173, −2.19796685980815850566575176915, −1.21325177578667372153513972808, −0.940193319874360338425704127048,
0.940193319874360338425704127048, 1.21325177578667372153513972808, 2.19796685980815850566575176915, 3.17748289583813502672208186173, 3.75030586513910626996503822543, 4.10097693297599374182001353842, 4.85296615641386915626758451398, 4.97038504677694939157559048920, 5.97665725329863433223916161740, 6.09965473708627903733411870985, 6.68738746488112880918032304464, 6.73788536351109334688842912040, 7.29678260320644648977236916626, 7.933233395447223951528066045261, 8.663013357009031343385744120378, 9.049635266734728437059057907380, 9.114235929380825439842956604773, 9.529679604819364368860346342313, 10.41028537391621499623050381498, 10.46033345850192554351588473598
