| L(s) = 1 | + 5·4-s + 4·7-s − 8·13-s + 9·16-s + 22·19-s + 38·25-s + 20·28-s + 64·31-s − 68·37-s − 122·43-s − 86·49-s − 40·52-s + 112·61-s − 35·64-s − 62·67-s + 130·73-s + 110·76-s + 76·79-s − 32·91-s − 230·97-s + 190·100-s − 80·103-s − 104·109-s + 36·112-s + 239·121-s + 320·124-s + 127-s + ⋯ |
| L(s) = 1 | + 5/4·4-s + 4/7·7-s − 0.615·13-s + 9/16·16-s + 1.15·19-s + 1.51·25-s + 5/7·28-s + 2.06·31-s − 1.83·37-s − 2.83·43-s − 1.75·49-s − 0.769·52-s + 1.83·61-s − 0.546·64-s − 0.925·67-s + 1.78·73-s + 1.44·76-s + 0.962·79-s − 0.351·91-s − 2.37·97-s + 1.89·100-s − 0.776·103-s − 0.954·109-s + 9/28·112-s + 1.97·121-s + 2.58·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.046346365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046346365\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 38 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 239 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 346 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3215 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2066 T^{2} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4439 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 115 T + p^{2} T^{2} )^{2} \) |
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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
−14.55966552129171568996663950980, −13.74604169606526625164836059246, −13.51448435198373463008825810544, −12.54067656517139737040194692719, −12.06995674983017770374194116785, −11.72877688759363686535417870674, −11.16580948122811582814913551340, −10.67495667156671470596313301077, −9.968739564659839038337420143615, −9.600538665297463033232587871654, −8.377248914493064909821380960478, −8.301424638954119487778952888415, −7.28003725026865185845865828635, −6.82388651889356636195690988481, −6.37807969982557596340368666874, −5.12602344708600068734670111311, −4.91767619016850799903901619266, −3.42511858789640607936777590708, −2.65536699747951785015437441508, −1.50473706047180571496209978742,
1.50473706047180571496209978742, 2.65536699747951785015437441508, 3.42511858789640607936777590708, 4.91767619016850799903901619266, 5.12602344708600068734670111311, 6.37807969982557596340368666874, 6.82388651889356636195690988481, 7.28003725026865185845865828635, 8.301424638954119487778952888415, 8.377248914493064909821380960478, 9.600538665297463033232587871654, 9.968739564659839038337420143615, 10.67495667156671470596313301077, 11.16580948122811582814913551340, 11.72877688759363686535417870674, 12.06995674983017770374194116785, 12.54067656517139737040194692719, 13.51448435198373463008825810544, 13.74604169606526625164836059246, 14.55966552129171568996663950980
