| L(s) = 1 | − 4·4-s − 5-s − 5·9-s − 8·11-s + 12·16-s − 2·19-s + 4·20-s + 25-s − 20·29-s − 2·31-s + 20·36-s − 6·41-s + 32·44-s + 5·45-s − 14·49-s + 8·55-s + 22·59-s − 24·61-s − 32·64-s + 18·71-s + 8·76-s − 20·79-s − 12·80-s + 16·81-s + 2·95-s + 40·99-s − 4·100-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.447·5-s − 5/3·9-s − 2.41·11-s + 3·16-s − 0.458·19-s + 0.894·20-s + 1/5·25-s − 3.71·29-s − 0.359·31-s + 10/3·36-s − 0.937·41-s + 4.82·44-s + 0.745·45-s − 2·49-s + 1.07·55-s + 2.86·59-s − 3.07·61-s − 4·64-s + 2.13·71-s + 0.917·76-s − 2.25·79-s − 1.34·80-s + 16/9·81-s + 0.205·95-s + 4.02·99-s − 2/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( 1 + T \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.872230948100296054595551024133, −8.347447740337488374596011500683, −8.232098142144935795769541453640, −7.66390102796328495298106373701, −7.31046477239498505884334187868, −6.12144203533511444176295483306, −5.46732296641201331380575932325, −5.44890211749766646861026772927, −4.92691899597547312483394760424, −4.20236737427813552615455555251, −3.44059941664901406912698236285, −3.14330660201336049402162930576, −2.08237608620337746445678971861, 0, 0,
2.08237608620337746445678971861, 3.14330660201336049402162930576, 3.44059941664901406912698236285, 4.20236737427813552615455555251, 4.92691899597547312483394760424, 5.44890211749766646861026772927, 5.46732296641201331380575932325, 6.12144203533511444176295483306, 7.31046477239498505884334187868, 7.66390102796328495298106373701, 8.232098142144935795769541453640, 8.347447740337488374596011500683, 8.872230948100296054595551024133
