| L(s) = 1 | + 2-s − 6·7-s + 2·8-s − 5·9-s + 2·11-s + 2·13-s − 6·14-s + 3·16-s + 6·17-s − 5·18-s + 4·19-s + 2·22-s − 4·23-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 + 3·41-s + 4·43-s − 4·46-s + 7·49-s − 6·53-s + 2·54-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.26·7-s + 0.707·8-s − 5/3·9-s + 0.603·11-s + 0.554·13-s − 1.60·14-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 + 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 + 0.468·41-s + 0.609·43-s − 0.589·46-s + 49-s − 0.824·53-s + 0.272·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.845406112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.845406112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 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 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 6 T + 29 T^{2} + 86 T^{3} + 29 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−9.123625028728332741430687360847, −8.490758687040857441096456548476, −8.055765449148048645102715813892, −8.052340763052009098380499226027, −7.76794448326395054663116292003, −7.66458340483034586153232499481, −6.85641006548110643489857091957, −6.79468986514786644034269216904, −6.29519055956427649636701945995, −6.25457145783786194151800388376, −6.14481878034005553832886128740, −5.67752709976573814465371046018, −5.34434896413398144079330404916, −5.10618202840430853759675585758, −4.69265989264030774649342414043, −4.33885853977740724054042198023, −3.84242103833045669188331486238, −3.63420380008328464383716876845, −3.30675513743247324004795574308, −3.12926557046848411375816244040, −2.62724458680717044781630426773, −2.46987239414356007730997068379, −1.62527961229887673978791818394, −0.941499314275310507191160679115, −0.57798830134644922398230629753,
0.57798830134644922398230629753, 0.941499314275310507191160679115, 1.62527961229887673978791818394, 2.46987239414356007730997068379, 2.62724458680717044781630426773, 3.12926557046848411375816244040, 3.30675513743247324004795574308, 3.63420380008328464383716876845, 3.84242103833045669188331486238, 4.33885853977740724054042198023, 4.69265989264030774649342414043, 5.10618202840430853759675585758, 5.34434896413398144079330404916, 5.67752709976573814465371046018, 6.14481878034005553832886128740, 6.25457145783786194151800388376, 6.29519055956427649636701945995, 6.79468986514786644034269216904, 6.85641006548110643489857091957, 7.66458340483034586153232499481, 7.76794448326395054663116292003, 8.052340763052009098380499226027, 8.055765449148048645102715813892, 8.490758687040857441096456548476, 9.123625028728332741430687360847
