| L(s) = 1 | − 3-s − 7-s − 2·9-s − 5·13-s − 5·17-s − 3·19-s + 21-s − 23-s + 3·27-s + 17·29-s − 2·31-s − 8·37-s + 5·39-s + 6·41-s − 10·43-s + 4·47-s − 8·49-s + 5·51-s − 5·53-s + 3·57-s − 15·59-s + 8·61-s + 2·63-s + 3·67-s + 69-s + 4·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.38·13-s − 1.21·17-s − 0.688·19-s + 0.218·21-s − 0.208·23-s + 0.577·27-s + 3.15·29-s − 0.359·31-s − 1.31·37-s + 0.800·39-s + 0.937·41-s − 1.52·43-s + 0.583·47-s − 8/7·49-s + 0.700·51-s − 0.686·53-s + 0.397·57-s − 1.95·59-s + 1.02·61-s + 0.251·63-s + 0.366·67-s + 0.120·69-s + 0.474·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} - 2 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 41 T^{2} + 126 T^{3} + 41 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 47 T^{2} + 166 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 37 T^{2} - 18 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 17 T + 171 T^{2} - 1110 T^{3} + 171 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 45 T^{2} - 4 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 137 T^{2} + 828 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 504 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 5 T + 117 T^{2} + 582 T^{3} + 117 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 15 T + 149 T^{2} + 986 T^{3} + 149 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 179 T^{2} - 912 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 23 T^{2} + 650 T^{3} + 23 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 456 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 119 T^{2} + 10 p T^{3} + 119 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 213 T^{2} + 284 T^{3} + 213 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 233 T^{2} + 1628 T^{3} + 233 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 215 T^{2} - 884 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 187 T^{2} - 1072 T^{3} + 187 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.15678963221824327705645482122, −7.02115491643811668755505906226, −6.77006631616680578447401808939, −6.63393319930827593956001374053, −6.35418621277178760598698345697, −6.23473370539326912187049822096, −6.03080033178334555285032181220, −5.63054429410986928929590013270, −5.35389767783948633406017925755, −5.25278228255720147298262288875, −4.87905151762228197892268743508, −4.65604440154645025376949627826, −4.60425357520523261133766822538, −4.32369137673830946908915165776, −3.98185020260172819393597062972, −3.68206496406990390320980561183, −3.30257415323498564664133562904, −3.02606685603409534858741777137, −2.90972891587859250073225639578, −2.48630991622693237153508007462, −2.28824360333817447398862658998, −2.13581359086509507455752262909, −1.53106951297946694612886954666, −1.25953533091134235073599248748, −0.899223550878336406223681041434, 0, 0, 0,
0.899223550878336406223681041434, 1.25953533091134235073599248748, 1.53106951297946694612886954666, 2.13581359086509507455752262909, 2.28824360333817447398862658998, 2.48630991622693237153508007462, 2.90972891587859250073225639578, 3.02606685603409534858741777137, 3.30257415323498564664133562904, 3.68206496406990390320980561183, 3.98185020260172819393597062972, 4.32369137673830946908915165776, 4.60425357520523261133766822538, 4.65604440154645025376949627826, 4.87905151762228197892268743508, 5.25278228255720147298262288875, 5.35389767783948633406017925755, 5.63054429410986928929590013270, 6.03080033178334555285032181220, 6.23473370539326912187049822096, 6.35418621277178760598698345697, 6.63393319930827593956001374053, 6.77006631616680578447401808939, 7.02115491643811668755505906226, 7.15678963221824327705645482122
