| L(s) = 1 | − 2·3-s − 6·5-s + 7-s + 9-s + 12·15-s − 10·17-s − 2·21-s + 18·25-s + 4·27-s − 6·35-s − 8·37-s − 2·41-s + 4·43-s − 6·45-s + 8·47-s + 49-s + 20·51-s + 12·59-s + 63-s − 20·67-s − 36·75-s + 4·79-s − 11·81-s − 12·83-s + 60·85-s − 2·89-s − 22·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.68·5-s + 0.377·7-s + 1/3·9-s + 3.09·15-s − 2.42·17-s − 0.436·21-s + 18/5·25-s + 0.769·27-s − 1.01·35-s − 1.31·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 2.80·51-s + 1.56·59-s + 0.125·63-s − 2.44·67-s − 4.15·75-s + 0.450·79-s − 1.22·81-s − 1.31·83-s + 6.50·85-s − 0.211·89-s − 2.18·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.84043029808953229621730651343, −7.18605041978648171160203518290, −7.03152579204446796229783936674, −6.65476613742299124786429917859, −6.04452734320831153928766356368, −5.39347511712501153185020299078, −4.98520525016032227931433692484, −4.39054590423470438150756628633, −4.14031154281970387253670453748, −3.83942205821574705570254421036, −2.99125426616399191829127973053, −2.37820478746382051308419156387, −1.22167662873155421266622221515, 0, 0,
1.22167662873155421266622221515, 2.37820478746382051308419156387, 2.99125426616399191829127973053, 3.83942205821574705570254421036, 4.14031154281970387253670453748, 4.39054590423470438150756628633, 4.98520525016032227931433692484, 5.39347511712501153185020299078, 6.04452734320831153928766356368, 6.65476613742299124786429917859, 7.03152579204446796229783936674, 7.18605041978648171160203518290, 7.84043029808953229621730651343
