| L(s) = 1 | − 3-s + 3·7-s − 2·9-s − 7·11-s − 5·13-s + 17-s − 4·19-s − 3·21-s − 6·23-s − 3·27-s + 3·29-s − 10·31-s + 7·33-s − 18·37-s + 5·39-s + 18·41-s − 9·47-s + 6·49-s − 51-s − 10·53-s + 4·57-s − 6·59-s + 24·61-s − 6·63-s − 10·67-s + 6·69-s − 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s − 2.11·11-s − 1.38·13-s + 0.242·17-s − 0.917·19-s − 0.654·21-s − 1.25·23-s − 0.577·27-s + 0.557·29-s − 1.79·31-s + 1.21·33-s − 2.95·37-s + 0.800·39-s + 2.81·41-s − 1.31·47-s + 6/7·49-s − 0.140·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s + 3.07·61-s − 0.755·63-s − 1.22·67-s + 0.722·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 7 T + 41 T^{2} + 146 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 24 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 27 T^{2} - 54 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 43 T^{2} + 160 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 101 T^{2} + 540 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 18 T + 191 T^{2} - 1388 T^{3} + 191 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 89 T^{2} + 64 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 93 T^{2} + 614 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 115 T^{2} + 588 T^{3} + 115 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 365 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 137 T^{2} + 828 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 193 T^{2} + 504 T^{3} + 193 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 740 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 17 T + 205 T^{2} + 2138 T^{3} + 205 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 107 T^{2} + 1168 T^{3} + 107 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 95 T^{2} - 464 T^{3} + 95 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 275 T^{2} - 1702 T^{3} + 275 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.082876127089380189839832886101, −7.87790200105060877477348006969, −7.60399754753964800460903854547, −7.54542987908597099670012899687, −7.32735822110112478313261256826, −6.89434365210721606998961765898, −6.81880127322729294539909910840, −6.10348725376720644281710701552, −6.09137515937161938248154834225, −5.89840781748521806540809136499, −5.38173999468615024240779896292, −5.29647862409811040104729372592, −5.24162928848717109250391356637, −4.80442548646629169240354802061, −4.66150121333856572645299577787, −4.28015894340489276023049710463, −3.89191139134345240411957788052, −3.62194843810101316545876391441, −3.33867624219640833089852111632, −2.77650583485363359895207542633, −2.45625013958186199906425101237, −2.37635761817996584566217216825, −2.01180528058163351252762877121, −1.58852306503842336278131521838, −1.17724760321238397859611137033, 0, 0, 0,
1.17724760321238397859611137033, 1.58852306503842336278131521838, 2.01180528058163351252762877121, 2.37635761817996584566217216825, 2.45625013958186199906425101237, 2.77650583485363359895207542633, 3.33867624219640833089852111632, 3.62194843810101316545876391441, 3.89191139134345240411957788052, 4.28015894340489276023049710463, 4.66150121333856572645299577787, 4.80442548646629169240354802061, 5.24162928848717109250391356637, 5.29647862409811040104729372592, 5.38173999468615024240779896292, 5.89840781748521806540809136499, 6.09137515937161938248154834225, 6.10348725376720644281710701552, 6.81880127322729294539909910840, 6.89434365210721606998961765898, 7.32735822110112478313261256826, 7.54542987908597099670012899687, 7.60399754753964800460903854547, 7.87790200105060877477348006969, 8.082876127089380189839832886101
