| 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
| \(L(2)\) |
\(\approx\) |
\(11.69885000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.69885000\) |
| \(L(\frac{5}{2})\) |
|
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
−8.202386346209289526231670610837, −7.71403795796236958248784325613, −7.67991425127547893280928663492, −7.51409169073608589497013281279, −7.20540338168408859510844235800, −6.95898037267198027139292401303, −6.45912133314122941848510778611, −6.19001210137120335715528945150, −6.15101624157999298709772207385, −5.79345129160459341820370928727, −5.13712063319873679911694728953, −5.03270524524780154303877929072, −4.63533354893557106222169469271, −4.15928995748761590860461091188, −3.88436153994345296560127776399, −3.71490406113758378223680575016, −3.36298204210559168350089038532, −3.11533064167192800063236362070, −3.08106392096691873655917663246, −2.11658343463963416487881973699, −2.09781342175238847082605197448, −1.90091444702013226902643315671, −1.09246965344094179948130688139, −0.68196056061167292137682808926, −0.55538034386562708055031170439,
0.55538034386562708055031170439, 0.68196056061167292137682808926, 1.09246965344094179948130688139, 1.90091444702013226902643315671, 2.09781342175238847082605197448, 2.11658343463963416487881973699, 3.08106392096691873655917663246, 3.11533064167192800063236362070, 3.36298204210559168350089038532, 3.71490406113758378223680575016, 3.88436153994345296560127776399, 4.15928995748761590860461091188, 4.63533354893557106222169469271, 5.03270524524780154303877929072, 5.13712063319873679911694728953, 5.79345129160459341820370928727, 6.15101624157999298709772207385, 6.19001210137120335715528945150, 6.45912133314122941848510778611, 6.95898037267198027139292401303, 7.20540338168408859510844235800, 7.51409169073608589497013281279, 7.67991425127547893280928663492, 7.71403795796236958248784325613, 8.202386346209289526231670610837
