| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s − 8·11-s + 12-s − 4·13-s − 16-s − 18-s + 8·22-s − 3·24-s + 25-s + 4·26-s − 27-s − 5·32-s + 8·33-s − 36-s − 20·37-s + 4·39-s + 8·44-s + 16·47-s + 48-s − 14·49-s − 50-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 2.41·11-s + 0.288·12-s − 1.10·13-s − 1/4·16-s − 0.235·18-s + 1.70·22-s − 0.612·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.883·32-s + 1.39·33-s − 1/6·36-s − 3.28·37-s + 0.640·39-s + 1.20·44-s + 2.33·47-s + 0.144·48-s − 2·49-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−10.67892245123374144028858824682, −10.60596946608548399795415271892, −10.26700477708582759405639390127, −9.523451675812284265792380668213, −8.982199030019329801304860528684, −8.270140448885357284476919824336, −7.66488013441745380243230842523, −7.40264879408133961073539513214, −6.55157891837643531715592264808, −5.32469243442401148699352691061, −5.23920392624592057055772361749, −4.54153907278640853760279816056, −3.29035819756263414840801407793, −2.11935007719153584295308930473, 0,
2.11935007719153584295308930473, 3.29035819756263414840801407793, 4.54153907278640853760279816056, 5.23920392624592057055772361749, 5.32469243442401148699352691061, 6.55157891837643531715592264808, 7.40264879408133961073539513214, 7.66488013441745380243230842523, 8.270140448885357284476919824336, 8.982199030019329801304860528684, 9.523451675812284265792380668213, 10.26700477708582759405639390127, 10.60596946608548399795415271892, 10.67892245123374144028858824682
