| L(s) = 1 | + 3-s − 3·5-s + 5·7-s − 2·9-s + 4·11-s + 5·13-s − 3·15-s − 11·17-s + 3·19-s + 5·21-s + 9·23-s + 6·25-s + 3·27-s + 3·29-s + 14·31-s + 4·33-s − 15·35-s + 14·37-s + 5·39-s − 10·41-s + 10·43-s + 6·45-s + 4·49-s − 11·51-s − 7·53-s − 12·55-s + 3·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.88·7-s − 2/3·9-s + 1.20·11-s + 1.38·13-s − 0.774·15-s − 2.66·17-s + 0.688·19-s + 1.09·21-s + 1.87·23-s + 6/5·25-s + 0.577·27-s + 0.557·29-s + 2.51·31-s + 0.696·33-s − 2.53·35-s + 2.30·37-s + 0.800·39-s − 1.56·41-s + 1.52·43-s + 0.894·45-s + 4/7·49-s − 1.54·51-s − 0.961·53-s − 1.61·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.345150793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.345150793\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 8 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 62 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 24 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 17 T^{2} - 24 T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 11 T + 83 T^{2} + 394 T^{3} + 83 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 53 T^{2} - 254 T^{3} + 53 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 125 T^{2} - 804 T^{3} + 125 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 157 T^{2} - 1016 T^{3} + 157 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 79 T^{2} + 348 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 12 T^{3} + 41 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 101 T^{2} - 64 T^{3} + 101 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 7 T + 145 T^{2} + 744 T^{3} + 145 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 7 T + 185 T^{2} - 818 T^{3} + 185 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 20 T + 215 T^{2} - 1800 T^{3} + 215 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 664 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 133 T^{2} - 624 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 13 T + 155 T^{2} + 1398 T^{3} + 155 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 181 T^{2} + 2228 T^{3} + 181 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 26 T + 441 T^{2} - 4668 T^{3} + 441 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3244 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 1144 T^{3} + 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.571804636702404216864940377154, −8.019530542442729921635327995281, −7.960933303078214418742344502371, −7.927434188684470920826000445948, −7.46480571247806026777822324476, −6.91267215541552086867782300793, −6.76846619580337473901910700630, −6.49793365073257769129074264248, −6.46217395749173829786352316714, −6.12944158579867901226943604447, −5.42674283011219229946173644213, −5.19173351179998773516836582147, −4.99471434604086188368997042762, −4.56381929450625266197745502837, −4.30015919988280673667498432821, −4.27712431237261146028795606149, −3.93547274482722403538255479999, −3.31161230351380995223024530359, −3.19299828891731030066194010586, −2.61295929268592052051967635369, −2.54729615123918031800923671531, −1.98188312223055493445124120904, −1.34524408198655258851606060564, −0.963802058085393332622397234047, −0.75172710569529784104943551369,
0.75172710569529784104943551369, 0.963802058085393332622397234047, 1.34524408198655258851606060564, 1.98188312223055493445124120904, 2.54729615123918031800923671531, 2.61295929268592052051967635369, 3.19299828891731030066194010586, 3.31161230351380995223024530359, 3.93547274482722403538255479999, 4.27712431237261146028795606149, 4.30015919988280673667498432821, 4.56381929450625266197745502837, 4.99471434604086188368997042762, 5.19173351179998773516836582147, 5.42674283011219229946173644213, 6.12944158579867901226943604447, 6.46217395749173829786352316714, 6.49793365073257769129074264248, 6.76846619580337473901910700630, 6.91267215541552086867782300793, 7.46480571247806026777822324476, 7.927434188684470920826000445948, 7.960933303078214418742344502371, 8.019530542442729921635327995281, 8.571804636702404216864940377154
