| L(s) = 1 | − 2-s + 2·5-s − 2·8-s − 5·9-s − 2·10-s + 2·11-s + 2·13-s + 3·16-s + 6·17-s + 5·18-s − 4·19-s − 2·22-s + 4·23-s − 7·25-s − 2·26-s + 2·27-s − 6·29-s − 16·31-s − 3·32-s − 6·34-s − 6·37-s + 4·38-s − 4·40-s − 3·41-s − 4·43-s − 10·45-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.894·5-s − 0.707·8-s − 5/3·9-s − 0.632·10-s + 0.603·11-s + 0.554·13-s + 3/4·16-s + 1.45·17-s + 1.17·18-s − 0.917·19-s − 0.426·22-s + 0.834·23-s − 7/5·25-s − 0.392·26-s + 0.384·27-s − 1.11·29-s − 2.87·31-s − 0.530·32-s − 1.02·34-s − 0.986·37-s + 0.648·38-s − 0.632·40-s − 0.468·41-s − 0.609·43-s − 1.49·45-s − 0.589·46-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
| \(L(1)\) |
\(\approx\) |
\(1.332172203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332172203\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 11 T^{2} - 16 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} + 6 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 162 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} - 216 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 308 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 16 T + 157 T^{2} + 1024 T^{3} + 157 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 336 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 121 T^{2} + 328 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 502 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 155 T^{2} - 628 T^{3} + 155 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 161 T^{2} - 784 T^{3} + 161 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 131 T^{2} + 60 T^{3} + 131 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 181 T^{2} + 218 T^{3} + 181 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 20 T + 297 T^{2} - 2706 T^{3} + 297 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 39 T^{2} - 536 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 32 T + 565 T^{2} - 6146 T^{3} + 565 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 185 T^{2} + 128 T^{3} + 185 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 119 T^{2} - 148 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 239 T^{2} + 916 T^{3} + 239 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.289199612632724593454937217380, −8.002216072546162513079179792019, −7.66810534253625841092538402831, −7.43814526007006941318885866566, −6.99459000058918531845381405610, −6.74131009681562026066490622436, −6.57430314484302818227185866074, −6.30643277305009937397688561806, −5.77909202695423802671379025481, −5.61624614602122726892334583331, −5.61482120724748311616575174308, −5.29283568276806955435739101562, −5.19745872601020281025378479611, −4.57646766569118575303268411598, −3.95670683955901499917894232290, −3.88723192902818605289598667715, −3.45485797790592698971423116814, −3.34624071936496934380600665338, −3.06438516134324274589512965699, −2.45364016747083893020545977343, −2.11175839060137877080805317388, −1.71390561615756558494992594435, −1.67855226256675456017819795986, −0.65278140707133254005518801526, −0.43859567533715214326457897958,
0.43859567533715214326457897958, 0.65278140707133254005518801526, 1.67855226256675456017819795986, 1.71390561615756558494992594435, 2.11175839060137877080805317388, 2.45364016747083893020545977343, 3.06438516134324274589512965699, 3.34624071936496934380600665338, 3.45485797790592698971423116814, 3.88723192902818605289598667715, 3.95670683955901499917894232290, 4.57646766569118575303268411598, 5.19745872601020281025378479611, 5.29283568276806955435739101562, 5.61482120724748311616575174308, 5.61624614602122726892334583331, 5.77909202695423802671379025481, 6.30643277305009937397688561806, 6.57430314484302818227185866074, 6.74131009681562026066490622436, 6.99459000058918531845381405610, 7.43814526007006941318885866566, 7.66810534253625841092538402831, 8.002216072546162513079179792019, 8.289199612632724593454937217380
