| 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
−8.655074250265874594654667643843, −8.114757177440921419069207299350, −8.045639809832312220968668237407, −7.46678658042041033049343800975, −6.77287251257420576381216593643, −6.66175602749173699922625283942, −5.96391714742476427877734512505, −5.31508624669203565794999838972, −4.48018036005395779311676580836, −4.12503217255581132297434015747, −3.50877292142339574199218689538, −2.63519340777820009043746148182, −1.90319955654708104595670502735, −1.40715345955998479729965074286, 0,
1.40715345955998479729965074286, 1.90319955654708104595670502735, 2.63519340777820009043746148182, 3.50877292142339574199218689538, 4.12503217255581132297434015747, 4.48018036005395779311676580836, 5.31508624669203565794999838972, 5.96391714742476427877734512505, 6.66175602749173699922625283942, 6.77287251257420576381216593643, 7.46678658042041033049343800975, 8.045639809832312220968668237407, 8.114757177440921419069207299350, 8.655074250265874594654667643843
