L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 4·13-s − 16-s − 4·17-s + 2·20-s + 3·25-s − 4·26-s + 4·29-s + 5·32-s − 4·34-s − 20·37-s + 6·40-s − 20·41-s − 14·49-s + 3·50-s + 4·52-s + 20·53-s + 4·58-s − 4·61-s + 7·64-s + 8·65-s + 4·68-s + 20·73-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.447·20-s + 3/5·25-s − 0.784·26-s + 0.742·29-s + 0.883·32-s − 0.685·34-s − 3.28·37-s + 0.948·40-s − 3.12·41-s − 2·49-s + 0.424·50-s + 0.554·52-s + 2.74·53-s + 0.525·58-s − 0.512·61-s + 7/8·64-s + 0.992·65-s + 0.485·68-s + 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−10.26700477708582759405639390127, −9.738851148469479131038939478590, −8.982199030019329801304860528684, −8.554588868470699894187777077269, −8.270140448885357284476919824336, −7.40264879408133961073539513214, −6.76955885158912340897756950945, −6.55157891837643531715592264808, −5.32469243442401148699352691061, −5.04860090093668311608273372623, −4.54153907278640853760279816056, −3.63412818236731287070311550143, −3.29035819756263414840801407793, −2.11935007719153584295308930473, 0,
2.11935007719153584295308930473, 3.29035819756263414840801407793, 3.63412818236731287070311550143, 4.54153907278640853760279816056, 5.04860090093668311608273372623, 5.32469243442401148699352691061, 6.55157891837643531715592264808, 6.76955885158912340897756950945, 7.40264879408133961073539513214, 8.270140448885357284476919824336, 8.554588868470699894187777077269, 8.982199030019329801304860528684, 9.738851148469479131038939478590, 10.26700477708582759405639390127