| L(s) = 1 | − 9·3-s − 6·5-s + 21·7-s + 54·9-s − 48·11-s − 6·13-s + 54·15-s + 54·17-s − 84·19-s − 189·21-s + 48·23-s − 123·25-s − 270·27-s − 18·29-s + 72·31-s + 432·33-s − 126·35-s − 210·37-s + 54·39-s + 414·41-s − 168·43-s − 324·45-s + 72·47-s + 294·49-s − 486·51-s − 402·53-s + 288·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.536·5-s + 1.13·7-s + 2·9-s − 1.31·11-s − 0.128·13-s + 0.929·15-s + 0.770·17-s − 1.01·19-s − 1.96·21-s + 0.435·23-s − 0.983·25-s − 1.92·27-s − 0.115·29-s + 0.417·31-s + 2.27·33-s − 0.608·35-s − 0.933·37-s + 0.221·39-s + 1.57·41-s − 0.595·43-s − 1.07·45-s + 0.223·47-s + 6/7·49-s − 1.33·51-s − 1.04·53-s + 0.706·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.016065487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.016065487\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 6 T + 159 T^{2} - 132 T^{3} + 159 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 48 T + 933 T^{2} + 32288 T^{3} + 933 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T - 165 T^{2} + 90020 T^{3} - 165 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 54 T + 699 p T^{2} - 505868 T^{3} + 699 p^{4} T^{4} - 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 84 T + 18705 T^{2} + 952056 T^{3} + 18705 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 48 T + 32001 T^{2} - 980384 T^{3} + 32001 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 18 T + 16059 T^{2} + 5726092 T^{3} + 16059 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 72 T + 59229 T^{2} - 1946992 T^{3} + 59229 p^{3} T^{4} - 72 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 210 T + 159891 T^{2} + 573532 p T^{3} + 159891 p^{3} T^{4} + 210 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 414 T + 225219 T^{2} - 52108924 T^{3} + 225219 p^{3} T^{4} - 414 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 168 T + 220857 T^{2} + 25991408 T^{3} + 220857 p^{3} T^{4} + 168 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 72 T + 159069 T^{2} - 32075248 T^{3} + 159069 p^{3} T^{4} - 72 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 402 T + 439587 T^{2} + 111857036 T^{3} + 439587 p^{3} T^{4} + 402 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 540 T + 384105 T^{2} + 233348008 T^{3} + 384105 p^{3} T^{4} + 540 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 798 T + 496155 T^{2} + 192224404 T^{3} + 496155 p^{3} T^{4} + 798 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 48 T + 552513 T^{2} - 15173344 T^{3} + 552513 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 456 T + 1036977 T^{2} + 306768720 T^{3} + 1036977 p^{3} T^{4} + 456 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1230 T + 728103 T^{2} - 316731204 T^{3} + 728103 p^{3} T^{4} - 1230 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1368 T + 1704573 T^{2} + 1166236880 T^{3} + 1704573 p^{3} T^{4} + 1368 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 60 T + 1039761 T^{2} - 22097560 T^{3} + 1039761 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2742 T + 4361523 T^{2} - 4390387500 T^{3} + 4361523 p^{3} T^{4} - 2742 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1950 T + 3376767 T^{2} - 3609510148 T^{3} + 3376767 p^{3} T^{4} - 1950 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.118850014182878643020843661794, −7.63163956407498061379112636400, −7.61858822488759997929426058616, −7.60921433711674256541015613819, −7.03520231072557564711633614782, −6.81450695522110904115724389047, −6.38864260578475342474464262680, −6.04640333793175955032166746280, −5.97527793291704594261505559098, −5.68020407355251566642528021679, −5.19032669859092107079004437724, −5.02405277841662530527076626532, −4.94940228033923228263870931230, −4.37388072437792487790328252330, −4.30519222575532088163390635248, −4.03201929426590247338679710569, −3.44193212975405965004190683215, −2.98827386049188764259619307761, −2.92037590904263606967820347813, −2.05263355326379615545968833042, −1.83363086504664927059073372871, −1.70488634707735000763823103674, −0.884914300537558433883225134824, −0.63813019365216755117812031425, −0.25089271005070443125425440372,
0.25089271005070443125425440372, 0.63813019365216755117812031425, 0.884914300537558433883225134824, 1.70488634707735000763823103674, 1.83363086504664927059073372871, 2.05263355326379615545968833042, 2.92037590904263606967820347813, 2.98827386049188764259619307761, 3.44193212975405965004190683215, 4.03201929426590247338679710569, 4.30519222575532088163390635248, 4.37388072437792487790328252330, 4.94940228033923228263870931230, 5.02405277841662530527076626532, 5.19032669859092107079004437724, 5.68020407355251566642528021679, 5.97527793291704594261505559098, 6.04640333793175955032166746280, 6.38864260578475342474464262680, 6.81450695522110904115724389047, 7.03520231072557564711633614782, 7.60921433711674256541015613819, 7.61858822488759997929426058616, 7.63163956407498061379112636400, 8.118850014182878643020843661794
