| L(s) = 1 | − 6·9-s − 3·11-s + 3·13-s − 6·17-s + 3·19-s + 6·23-s − 27-s − 3·29-s − 15·31-s − 9·37-s − 9·41-s + 3·43-s + 3·47-s − 12·49-s + 3·53-s + 6·59-s − 9·61-s − 3·67-s − 9·71-s + 9·73-s − 12·79-s + 18·81-s + 9·83-s + 6·89-s − 21·97-s + 18·99-s − 12·101-s + ⋯ |
| L(s) = 1 | − 2·9-s − 0.904·11-s + 0.832·13-s − 1.45·17-s + 0.688·19-s + 1.25·23-s − 0.192·27-s − 0.557·29-s − 2.69·31-s − 1.47·37-s − 1.40·41-s + 0.457·43-s + 0.437·47-s − 1.71·49-s + 0.412·53-s + 0.781·59-s − 1.15·61-s − 0.366·67-s − 1.06·71-s + 1.05·73-s − 1.35·79-s + 2·81-s + 0.987·83-s + 0.635·89-s − 2.13·97-s + 1.80·99-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 12 T^{2} + 9 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 27 T^{2} + 49 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 95 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 205 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 187 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 225 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 15 T + 159 T^{2} + 1019 T^{3} + 159 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 9 T + 81 T^{2} + 17 p T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 111 T^{2} + 629 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 3 T + 120 T^{2} - 239 T^{3} + 120 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 299 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 114 T^{2} - 207 T^{3} + 114 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 412 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 865 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 3 T + 195 T^{2} + 385 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 192 T^{2} + 1097 T^{3} + 192 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 9 T + 237 T^{2} - 1323 T^{3} + 237 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 12 T + 258 T^{2} + 1825 T^{3} + 258 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 9 T + 147 T^{2} - 1205 T^{3} + 147 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 6 T + 258 T^{2} - 1017 T^{3} + 258 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 21 T + 381 T^{2} + 4181 T^{3} + 381 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.075090097932257964949754669774, −7.68709264636382567784982554121, −7.30515312906735016620445692058, −7.24777202094073446204374677016, −6.83811836414562978227246153356, −6.67446778875160054132559089883, −6.51812414607771200458582306400, −6.03976435627965018324782052911, −5.75172833104877227755021403517, −5.73404515996357486166852052872, −5.31751480890423155789919971311, −5.11518292965691443897873125193, −5.09305823951035901966876970411, −4.70132888224051495653190760036, −4.27372445316006002731419443181, −3.90131940486388605497069420213, −3.55421250105112568367958548171, −3.41840783933411250601966509292, −3.31085828372656333225823796987, −2.66880638982154988135143913954, −2.47998568446861182024042590298, −2.44176648847454841233427057372, −1.78364103203150793156084765108, −1.40581723816416527127611914855, −1.21292751933200864647673746255, 0, 0, 0,
1.21292751933200864647673746255, 1.40581723816416527127611914855, 1.78364103203150793156084765108, 2.44176648847454841233427057372, 2.47998568446861182024042590298, 2.66880638982154988135143913954, 3.31085828372656333225823796987, 3.41840783933411250601966509292, 3.55421250105112568367958548171, 3.90131940486388605497069420213, 4.27372445316006002731419443181, 4.70132888224051495653190760036, 5.09305823951035901966876970411, 5.11518292965691443897873125193, 5.31751480890423155789919971311, 5.73404515996357486166852052872, 5.75172833104877227755021403517, 6.03976435627965018324782052911, 6.51812414607771200458582306400, 6.67446778875160054132559089883, 6.83811836414562978227246153356, 7.24777202094073446204374677016, 7.30515312906735016620445692058, 7.68709264636382567784982554121, 8.075090097932257964949754669774
