| L(s) = 1 | − 8·4-s + 7-s − 2·13-s + 40·16-s − 19-s − 20·25-s − 8·28-s − 4·31-s + 11·37-s + 5·43-s + 7·49-s + 16·52-s − 13·61-s − 160·64-s − 11·67-s − 10·73-s + 8·76-s − 13·79-s − 2·91-s + 14·97-s + 160·100-s − 20·103-s − 17·109-s + 40·112-s − 44·121-s + 32·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4·4-s + 0.377·7-s − 0.554·13-s + 10·16-s − 0.229·19-s − 4·25-s − 1.51·28-s − 0.718·31-s + 1.80·37-s + 0.762·43-s + 49-s + 2.21·52-s − 1.66·61-s − 20·64-s − 1.34·67-s − 1.17·73-s + 0.917·76-s − 1.46·79-s − 0.209·91-s + 1.42·97-s + 16·100-s − 1.97·103-s − 1.62·109-s + 3.77·112-s − 4·121-s + 2.87·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 29 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_4\times C_2$ | \( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_4\times C_2$ | \( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 44 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_4\times C_2$ | \( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_4\times C_2$ | \( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} - 184 p T^{5} - 15 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 517 p T^{5} + 84 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_4\times C_2$ | \( 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} + 305 p T^{5} - 18 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_4\times C_2$ | \( 1 + 13 T + 108 T^{2} + 611 T^{3} + 1355 T^{4} + 611 p T^{5} + 108 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 + 11 T + 54 T^{2} - 143 T^{3} - 5191 T^{4} - 143 p T^{5} + 54 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 460 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 13 T + 90 T^{2} + 143 T^{3} - 5251 T^{4} + 143 p T^{5} + 90 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} - 28 p T^{5} + 99 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.80301203246656688854737903269, −5.67591118116113773684498184987, −5.44478910106616633733563200060, −5.34654806383379519053839353412, −5.30432906245248856915194150942, −4.77144654773534155492871763363, −4.69190740101543008274899999648, −4.58560557516145415607200296221, −4.37379820046605395706154618229, −4.22779291434716124932379545367, −4.06444897393581446474919392252, −4.02758453305681070578143329897, −3.92808481731856575411189618124, −3.44127997653737385813060965211, −3.20849347332780585268741605872, −3.19118721596011276153274623370, −3.18458874315738660889833243891, −2.47341979823624841074061692368, −2.41913107615580637458210270704, −2.10436212484739088754172804614, −1.91164743924746212893676156274, −1.39912475947805497258548745717, −1.28486526853184290613784953983, −1.00376856468987677843214071953, −0.932877276837166783780316361544, 0, 0, 0, 0,
0.932877276837166783780316361544, 1.00376856468987677843214071953, 1.28486526853184290613784953983, 1.39912475947805497258548745717, 1.91164743924746212893676156274, 2.10436212484739088754172804614, 2.41913107615580637458210270704, 2.47341979823624841074061692368, 3.18458874315738660889833243891, 3.19118721596011276153274623370, 3.20849347332780585268741605872, 3.44127997653737385813060965211, 3.92808481731856575411189618124, 4.02758453305681070578143329897, 4.06444897393581446474919392252, 4.22779291434716124932379545367, 4.37379820046605395706154618229, 4.58560557516145415607200296221, 4.69190740101543008274899999648, 4.77144654773534155492871763363, 5.30432906245248856915194150942, 5.34654806383379519053839353412, 5.44478910106616633733563200060, 5.67591118116113773684498184987, 5.80301203246656688854737903269
