| L(s) = 1 | − 4·3-s − 4·4-s − 2·5-s + 6·9-s − 2·11-s + 16·12-s + 8·15-s + 12·16-s + 8·20-s + 2·23-s − 7·25-s + 4·27-s − 18·31-s + 8·33-s − 24·36-s − 4·37-s + 8·44-s − 12·45-s + 14·47-s − 48·48-s − 10·49-s − 4·53-s + 4·55-s − 28·59-s − 32·60-s − 32·64-s + 4·67-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2·4-s − 0.894·5-s + 2·9-s − 0.603·11-s + 4.61·12-s + 2.06·15-s + 3·16-s + 1.78·20-s + 0.417·23-s − 7/5·25-s + 0.769·27-s − 3.23·31-s + 1.39·33-s − 4·36-s − 0.657·37-s + 1.20·44-s − 1.78·45-s + 2.04·47-s − 6.92·48-s − 1.42·49-s − 0.549·53-s + 0.539·55-s − 3.64·59-s − 4.13·60-s − 4·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1234321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1234321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 101 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.74187934910278185224762610896, −7.27123391412294223311092606979, −6.33628735398302806971209845546, −6.16055116457811093791815485098, −5.66222777548012373876842190687, −5.17330498735283860298501426044, −5.05190011683367889598807367642, −4.70898772624388144319899057533, −3.91187178475190732797803079948, −3.72933468340090913789145872057, −3.14223806423963647907216241485, −1.85866097915362029590487734313, −0.845638584611923894041993845993, 0, 0,
0.845638584611923894041993845993, 1.85866097915362029590487734313, 3.14223806423963647907216241485, 3.72933468340090913789145872057, 3.91187178475190732797803079948, 4.70898772624388144319899057533, 5.05190011683367889598807367642, 5.17330498735283860298501426044, 5.66222777548012373876842190687, 6.16055116457811093791815485098, 6.33628735398302806971209845546, 7.27123391412294223311092606979, 7.74187934910278185224762610896
