| L(s) = 1 | + 3·3-s + 2·7-s + 6·9-s + 4·11-s + 6·13-s + 4·17-s + 6·19-s + 6·21-s − 3·23-s + 25-s + 10·27-s − 2·29-s − 4·31-s + 12·33-s + 14·37-s + 18·39-s − 2·41-s − 2·43-s − 16·47-s − 5·49-s + 12·51-s − 4·53-s + 18·57-s − 12·59-s + 22·61-s + 12·63-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.755·7-s + 2·9-s + 1.20·11-s + 1.66·13-s + 0.970·17-s + 1.37·19-s + 1.30·21-s − 0.625·23-s + 1/5·25-s + 1.92·27-s − 0.371·29-s − 0.718·31-s + 2.08·33-s + 2.30·37-s + 2.88·39-s − 0.312·41-s − 0.304·43-s − 2.33·47-s − 5/7·49-s + 1.68·51-s − 0.549·53-s + 2.38·57-s − 1.56·59-s + 2.81·61-s + 1.51·63-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.625374497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.625374497\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 - T^{2} + 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 9 T^{2} - 20 T^{3} + 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} - 120 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 220 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 184 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 139 T^{2} - 884 T^{3} + 139 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 60 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 77 T^{2} - 12 T^{3} + 77 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 173 T^{2} + 1376 T^{3} + 173 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} - 168 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 161 T^{2} + 1352 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 22 T + 307 T^{2} - 2884 T^{3} + 307 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 149 T^{2} - 84 T^{3} + 149 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 85 T^{2} + 1392 T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 449 T^{2} + 4644 T^{3} + 449 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 504 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 227 T^{2} + 1120 T^{3} + 227 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3196 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.741158672248007240619437629905, −9.251210399008956117268990116946, −8.917382188033609842943386201798, −8.835206714784139475157934830658, −8.314857791562294143984530421266, −8.108130592088647416839368747514, −8.049204183291778142387603396240, −7.55969747885931562150524405727, −7.38744311685683997301004653824, −7.00304345149988642058209307039, −6.51881503811754765688253498091, −6.20019300546419628763333340518, −6.10774384929442049482515829042, −5.45420201330113245774286302838, −5.10473166142377301052699449770, −4.84425447278574149094917013672, −4.20327172086876473975540025475, −3.83865868387026356686091511439, −3.81422658709136073325009122003, −3.34314264910790591298518562710, −2.83154157020359856438247917147, −2.64136067879323748687121798124, −1.59750872432190142870419687390, −1.54850387854144576745246497437, −1.16847977518486291489816016315,
1.16847977518486291489816016315, 1.54850387854144576745246497437, 1.59750872432190142870419687390, 2.64136067879323748687121798124, 2.83154157020359856438247917147, 3.34314264910790591298518562710, 3.81422658709136073325009122003, 3.83865868387026356686091511439, 4.20327172086876473975540025475, 4.84425447278574149094917013672, 5.10473166142377301052699449770, 5.45420201330113245774286302838, 6.10774384929442049482515829042, 6.20019300546419628763333340518, 6.51881503811754765688253498091, 7.00304345149988642058209307039, 7.38744311685683997301004653824, 7.55969747885931562150524405727, 8.049204183291778142387603396240, 8.108130592088647416839368747514, 8.314857791562294143984530421266, 8.835206714784139475157934830658, 8.917382188033609842943386201798, 9.251210399008956117268990116946, 9.741158672248007240619437629905
