| L(s) = 1 | + 2·2-s + 2·4-s − 5·5-s + 2·8-s − 10·10-s + 4·11-s − 3·13-s − 4·17-s + 7·19-s − 10·20-s + 8·22-s − 23-s + 7·25-s − 6·26-s + 7·29-s − 3·31-s − 8·34-s − 10·37-s + 14·38-s − 10·40-s − 6·41-s + 9·43-s + 8·44-s − 2·46-s − 17·47-s + 14·50-s − 6·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 2.23·5-s + 0.707·8-s − 3.16·10-s + 1.20·11-s − 0.832·13-s − 0.970·17-s + 1.60·19-s − 2.23·20-s + 1.70·22-s − 0.208·23-s + 7/5·25-s − 1.17·26-s + 1.29·29-s − 0.538·31-s − 1.37·34-s − 1.64·37-s + 2.27·38-s − 1.58·40-s − 0.937·41-s + 1.37·43-s + 1.20·44-s − 0.294·46-s − 2.47·47-s + 1.97·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + p T + 18 T^{2} + 43 T^{3} + 18 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 86 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 122 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 54 T^{2} - 203 T^{3} + 54 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 28 T^{2} + 89 T^{3} + 28 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 74 T^{2} - 403 T^{3} + 74 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 52 T^{2} + 137 T^{3} + 52 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 119 T^{2} + 658 T^{3} + 119 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 17 T + 230 T^{2} + 1745 T^{3} + 230 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 13 T + 198 T^{2} + 1369 T^{3} + 198 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 22 T + 321 T^{2} + 2848 T^{3} + 321 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 343 T^{2} + 3152 T^{3} + 343 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 165 T^{2} + 1228 T^{3} + 165 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 169 T^{2} + 374 T^{3} + 169 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 5 T + 831 T^{3} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - T + 168 T^{2} - 257 T^{3} + 168 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 23 T + 376 T^{2} + 4021 T^{3} + 376 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 11 T + 212 T^{2} - 1979 T^{3} + 212 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 220 T^{2} + 575 T^{3} + 220 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.69098578794233273220300314708, −7.22712537660392855090900714318, −6.90839018769688655683602901641, −6.82902600790185049339474658731, −6.65928795203886038964882420826, −6.29734938509435267312377343293, −6.29275323792760032661943367003, −5.74658259521821364202068332874, −5.40190025609478708952139786399, −5.39734928446845248975801914006, −4.90166000724959548151575766403, −4.71130323537766228405095667017, −4.58447975218212735697029798582, −4.21554498710043364305299175627, −4.16911793389703319969757932736, −4.04376045414144264031216977836, −3.54038602456532247663705564982, −3.27350512316635912566914136701, −3.06809674577558386794148505998, −2.88348119257592294393866809695, −2.76531280423310507320491949518, −1.87848320816067707672919345844, −1.80770753951125917137428588561, −1.29643835651447510602143817061, −1.23855841890560397048925522725, 0, 0, 0,
1.23855841890560397048925522725, 1.29643835651447510602143817061, 1.80770753951125917137428588561, 1.87848320816067707672919345844, 2.76531280423310507320491949518, 2.88348119257592294393866809695, 3.06809674577558386794148505998, 3.27350512316635912566914136701, 3.54038602456532247663705564982, 4.04376045414144264031216977836, 4.16911793389703319969757932736, 4.21554498710043364305299175627, 4.58447975218212735697029798582, 4.71130323537766228405095667017, 4.90166000724959548151575766403, 5.39734928446845248975801914006, 5.40190025609478708952139786399, 5.74658259521821364202068332874, 6.29275323792760032661943367003, 6.29734938509435267312377343293, 6.65928795203886038964882420826, 6.82902600790185049339474658731, 6.90839018769688655683602901641, 7.22712537660392855090900714318, 7.69098578794233273220300314708
