| L(s) = 1 | + 2-s + 4-s − 6·5-s + 8-s − 6·10-s − 8·13-s + 16-s − 6·20-s + 17·25-s − 8·26-s − 12·29-s + 32-s + 4·37-s − 6·40-s + 12·41-s − 13·49-s + 17·50-s − 8·52-s − 18·53-s − 12·58-s + 16·61-s + 64-s + 48·65-s − 14·73-s + 4·74-s − 6·80-s + 12·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 2.68·5-s + 0.353·8-s − 1.89·10-s − 2.21·13-s + 1/4·16-s − 1.34·20-s + 17/5·25-s − 1.56·26-s − 2.22·29-s + 0.176·32-s + 0.657·37-s − 0.948·40-s + 1.87·41-s − 1.85·49-s + 2.40·50-s − 1.10·52-s − 2.47·53-s − 1.57·58-s + 2.04·61-s + 1/8·64-s + 5.95·65-s − 1.63·73-s + 0.464·74-s − 0.670·80-s + 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.86482571623536481838849135667, −9.881080480790549207963868930122, −9.512741326522107701681952133776, −8.737948409524417511387388417952, −7.82723843094092867969941089213, −7.61880171096277894570972400592, −7.51148950331839783385046384041, −6.71508725988629390136097161445, −5.85096393191491862373045737952, −4.77709333112371596512170551343, −4.70373764496519643229180296651, −3.81269975524617156560988745696, −3.38549178369413487864105189864, −2.36301005321257278434888317321, 0,
2.36301005321257278434888317321, 3.38549178369413487864105189864, 3.81269975524617156560988745696, 4.70373764496519643229180296651, 4.77709333112371596512170551343, 5.85096393191491862373045737952, 6.71508725988629390136097161445, 7.51148950331839783385046384041, 7.61880171096277894570972400592, 7.82723843094092867969941089213, 8.737948409524417511387388417952, 9.512741326522107701681952133776, 9.881080480790549207963868930122, 10.86482571623536481838849135667
