| L(s) = 1 | − 5-s + 3·7-s + 2·11-s + 3·13-s − 2·17-s − 3·19-s + 14·23-s − 10·25-s − 29-s + 3·31-s − 3·35-s − 3·37-s − 3·43-s + 21·47-s + 6·49-s − 6·53-s − 2·55-s + 31·59-s + 6·61-s − 3·65-s − 6·67-s + 17·71-s − 3·73-s + 6·77-s + 9·79-s + 20·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.603·11-s + 0.832·13-s − 0.485·17-s − 0.688·19-s + 2.91·23-s − 2·25-s − 0.185·29-s + 0.538·31-s − 0.507·35-s − 0.493·37-s − 0.457·43-s + 3.06·47-s + 6/7·49-s − 0.824·53-s − 0.269·55-s + 4.03·59-s + 0.768·61-s − 0.372·65-s − 0.733·67-s + 2.01·71-s − 0.351·73-s + 0.683·77-s + 1.01·79-s + 2.19·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.939529212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.939529212\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 11 T^{2} + 9 T^{3} + 11 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 8 T^{2} + 15 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 51 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 32 T^{2} + 21 T^{3} + 32 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 35 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 122 T^{2} - 675 T^{3} + 122 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 47 T^{2} - 51 T^{3} + 47 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 57 T^{2} - 159 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 81 T^{2} + 199 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 90 T^{2} - 9 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 539 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 21 T + 261 T^{2} - 2181 T^{3} + 261 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 162 T^{2} + 627 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 31 T + 485 T^{2} - 4647 T^{3} + 485 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T - 12 T^{2} + 357 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 186 T^{2} + 811 T^{3} + 186 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 17 T + 119 T^{2} - 507 T^{3} + 119 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 195 T^{2} + 359 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 9 T + 195 T^{2} - 1053 T^{3} + 195 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 362 T^{2} - 3447 T^{3} + 362 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T + 216 T^{2} + 1425 T^{3} + 216 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 147 T^{2} + 1673 T^{3} + 147 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.97782190832686351118338297681, −6.50066208125177032786157463633, −6.42466648685753167697175721979, −6.27699519968380166799767301530, −5.83113604901168522371293506516, −5.64336960870370262541422985991, −5.40886056400286974865248598815, −5.15273829036280375444361069052, −5.03960761632084296218662378952, −4.74788206005461895825708580499, −4.40110390335821917357085579978, −4.11323152629587696050110987547, −4.09782208467360353152472767827, −3.64054744971239021357903051434, −3.59045150923644519699240423266, −3.38813847878935540341216842100, −2.83704011555109870081223795959, −2.45095078448213124794594267885, −2.43711051284524830761699879553, −2.02211348189375012624095791969, −1.79416779975081240686605938251, −1.38188779077408310192678185846, −0.903328096917343370678510882771, −0.834459354795878086136421316391, −0.44877236400210296657002500635,
0.44877236400210296657002500635, 0.834459354795878086136421316391, 0.903328096917343370678510882771, 1.38188779077408310192678185846, 1.79416779975081240686605938251, 2.02211348189375012624095791969, 2.43711051284524830761699879553, 2.45095078448213124794594267885, 2.83704011555109870081223795959, 3.38813847878935540341216842100, 3.59045150923644519699240423266, 3.64054744971239021357903051434, 4.09782208467360353152472767827, 4.11323152629587696050110987547, 4.40110390335821917357085579978, 4.74788206005461895825708580499, 5.03960761632084296218662378952, 5.15273829036280375444361069052, 5.40886056400286974865248598815, 5.64336960870370262541422985991, 5.83113604901168522371293506516, 6.27699519968380166799767301530, 6.42466648685753167697175721979, 6.50066208125177032786157463633, 6.97782190832686351118338297681
