| L(s) = 1 | + 2·3-s + 4-s − 2·5-s − 3·9-s + 11-s + 2·12-s − 4·15-s + 16-s − 2·20-s − 12·23-s + 3·25-s − 14·27-s − 14·31-s + 2·33-s − 3·36-s − 14·37-s + 44-s + 6·45-s + 12·47-s + 2·48-s + 11·49-s − 6·53-s − 2·55-s − 12·59-s − 4·60-s + 64-s + 16·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.894·5-s − 9-s + 0.301·11-s + 0.577·12-s − 1.03·15-s + 1/4·16-s − 0.447·20-s − 2.50·23-s + 3/5·25-s − 2.69·27-s − 2.51·31-s + 0.348·33-s − 1/2·36-s − 2.30·37-s + 0.150·44-s + 0.894·45-s + 1.75·47-s + 0.288·48-s + 11/7·49-s − 0.824·53-s − 0.269·55-s − 1.56·59-s − 0.516·60-s + 1/8·64-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.996361573480265218453634288731, −8.544205197596580525949431141201, −8.285662434899332908926558240119, −7.56981196469027501153188760806, −7.50897098940210363365143353678, −6.80791132757835172680275047752, −5.97360770820880552844865151017, −5.73876439085829035805370640147, −5.06550696201468645859663862777, −4.01173135747517138761270457828, −3.64376558485075906520727529259, −3.32600209888818226598606709611, −2.30342494378677467481205459308, −1.97170842587956421114470880781, 0,
1.97170842587956421114470880781, 2.30342494378677467481205459308, 3.32600209888818226598606709611, 3.64376558485075906520727529259, 4.01173135747517138761270457828, 5.06550696201468645859663862777, 5.73876439085829035805370640147, 5.97360770820880552844865151017, 6.80791132757835172680275047752, 7.50897098940210363365143353678, 7.56981196469027501153188760806, 8.285662434899332908926558240119, 8.544205197596580525949431141201, 8.996361573480265218453634288731
