| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 4·11-s + 2·12-s + 2·13-s − 4·16-s − 2·18-s + 8·22-s − 12·23-s − 4·26-s + 27-s + 8·32-s − 4·33-s + 2·36-s + 4·37-s + 2·39-s − 8·44-s + 24·46-s − 4·47-s − 4·48-s − 5·49-s + 4·52-s − 2·54-s + 20·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.20·11-s + 0.577·12-s + 0.554·13-s − 16-s − 0.471·18-s + 1.70·22-s − 2.50·23-s − 0.784·26-s + 0.192·27-s + 1.41·32-s − 0.696·33-s + 1/3·36-s + 0.657·37-s + 0.320·39-s − 1.20·44-s + 3.53·46-s − 0.583·47-s − 0.577·48-s − 5/7·49-s + 0.554·52-s − 0.272·54-s + 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−8.686228257120952969207606021954, −8.134034752267247325522916900531, −7.965886472126962800720761127653, −7.57404393602003080106109524301, −6.96141726621690601574492077524, −6.47388999507393793454845833318, −5.87930303915993667052597404250, −5.34490029281998815763412767656, −4.63718444572594903228967418102, −4.01465426298836833468759021488, −3.50818980571584578672789924466, −2.44041792194616260799140408514, −2.23186561512245260821665501913, −1.24790523142424415923064013417, 0,
1.24790523142424415923064013417, 2.23186561512245260821665501913, 2.44041792194616260799140408514, 3.50818980571584578672789924466, 4.01465426298836833468759021488, 4.63718444572594903228967418102, 5.34490029281998815763412767656, 5.87930303915993667052597404250, 6.47388999507393793454845833318, 6.96141726621690601574492077524, 7.57404393602003080106109524301, 7.965886472126962800720761127653, 8.134034752267247325522916900531, 8.686228257120952969207606021954
