| L(s) = 1 | + 4·2-s − 5·3-s + 7·4-s − 2·5-s − 20·6-s + 7·8-s + 10·9-s − 8·10-s − 4·11-s − 35·12-s − 7·13-s + 10·15-s + 7·16-s − 3·17-s + 40·18-s + 11·19-s − 14·20-s − 16·22-s − 13·23-s − 35·24-s − 3·25-s − 28·26-s − 6·27-s + 2·29-s + 40·30-s + 14·32-s + 20·33-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.88·3-s + 7/2·4-s − 0.894·5-s − 8.16·6-s + 2.47·8-s + 10/3·9-s − 2.52·10-s − 1.20·11-s − 10.1·12-s − 1.94·13-s + 2.58·15-s + 7/4·16-s − 0.727·17-s + 9.42·18-s + 2.52·19-s − 3.13·20-s − 3.41·22-s − 2.71·23-s − 7.14·24-s − 3/5·25-s − 5.49·26-s − 1.15·27-s + 0.371·29-s + 7.30·30-s + 2.47·32-s + 3.48·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9379627764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9379627764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - p^{2} T + 9 T^{2} - 15 T^{3} + 9 p T^{4} - p^{4} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 5 T + 5 p T^{2} + 31 T^{3} + 5 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 80 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 7 T + 3 p T^{2} + 133 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 3 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 11 T + 81 T^{2} - 431 T^{3} + 81 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 13 T + 109 T^{2} + 627 T^{3} + 109 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_6$ | \( 1 - 2 T + 51 T^{2} - 108 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 9 T^{2} + 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 93 T^{2} - 195 T^{3} + 93 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 11 T + 125 T^{2} - 945 T^{3} + 125 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 13 T + 195 T^{2} + 1293 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 20 T + 283 T^{2} + 2352 T^{3} + 283 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 576 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 8 T + 139 T^{2} + 632 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 36 T + 605 T^{2} + 6160 T^{3} + 605 p T^{4} + 36 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 2 T + 177 T^{2} + 276 T^{3} + 177 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 2 T + 183 T^{2} - 284 T^{3} + 183 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 20 T + 333 T^{2} - 3224 T^{3} + 333 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 149 T^{2} - 4 p T^{3} + 149 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - T + 55 T^{2} - 1227 T^{3} + 55 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 11 T + 315 T^{2} + 2147 T^{3} + 315 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.85468553290404375724846619768, −7.59507898958865444637740430959, −7.48438010794385192673065596103, −7.42635240143945743714368258746, −6.74468152094969572763853178761, −6.39258486906303408168083059614, −6.22797098980346550870837753264, −6.14876340682585823398242266050, −5.80237862850420485691593078890, −5.51140958432480097000922858521, −5.23178742726866875793296207504, −5.11794426808117334530315993013, −5.08670923854805152945192932999, −4.49114217773277642118662473924, −4.48163669275998840225861502969, −4.32215626195390667952434150714, −3.89773426617328321283701608512, −3.41623856830810583787432527366, −3.15680495958965751677223592807, −2.76469789548241240077393575191, −2.67889034809801640955463109591, −1.71181003131902863440747123838, −1.66334054066544373135687737444, −0.43787611365830023183566987952, −0.36101717244637400149124311412,
0.36101717244637400149124311412, 0.43787611365830023183566987952, 1.66334054066544373135687737444, 1.71181003131902863440747123838, 2.67889034809801640955463109591, 2.76469789548241240077393575191, 3.15680495958965751677223592807, 3.41623856830810583787432527366, 3.89773426617328321283701608512, 4.32215626195390667952434150714, 4.48163669275998840225861502969, 4.49114217773277642118662473924, 5.08670923854805152945192932999, 5.11794426808117334530315993013, 5.23178742726866875793296207504, 5.51140958432480097000922858521, 5.80237862850420485691593078890, 6.14876340682585823398242266050, 6.22797098980346550870837753264, 6.39258486906303408168083059614, 6.74468152094969572763853178761, 7.42635240143945743714368258746, 7.48438010794385192673065596103, 7.59507898958865444637740430959, 7.85468553290404375724846619768
