| L(s) = 1 | − 6·2-s − 9·3-s + 24·4-s + 54·6-s − 21·7-s − 80·8-s + 54·9-s + 11·11-s − 216·12-s − 39·13-s + 126·14-s + 240·16-s − 23·17-s − 324·18-s − 38·19-s + 189·21-s − 66·22-s − 44·23-s + 720·24-s − 126·25-s + 234·26-s − 270·27-s − 504·28-s − 328·29-s + 159·31-s − 672·32-s − 99·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 3.67·6-s − 1.13·7-s − 3.53·8-s + 2·9-s + 0.301·11-s − 5.19·12-s − 0.832·13-s + 2.40·14-s + 15/4·16-s − 0.328·17-s − 4.24·18-s − 0.458·19-s + 1.96·21-s − 0.639·22-s − 0.398·23-s + 6.12·24-s − 1.00·25-s + 1.76·26-s − 1.92·27-s − 3.40·28-s − 2.10·29-s + 0.921·31-s − 3.71·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05970541230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05970541230\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 126 T^{2} - 392 T^{3} + 126 p^{3} T^{4} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - p T - 47 T^{2} + 60994 T^{3} - 47 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 23 T + 4725 T^{2} - 175138 T^{3} + 4725 p^{3} T^{4} + 23 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 p T + 18390 T^{2} + 466324 T^{3} + 18390 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 44 T + 19888 T^{2} + 376712 T^{3} + 19888 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 328 T + 55076 T^{2} + 6583854 T^{3} + 55076 p^{3} T^{4} + 328 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 159 T + 66465 T^{2} - 6228994 T^{3} + 66465 p^{3} T^{4} - 159 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 227 T + 104277 T^{2} + 12256054 T^{3} + 104277 p^{3} T^{4} + 227 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 196995 T^{2} + 998068 T^{3} + 196995 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 126950 T^{2} - 9127264 T^{3} + 126950 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 309 T + 199875 T^{2} + 23558 p^{2} T^{3} + 199875 p^{3} T^{4} + 309 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 849 T + 655563 T^{2} + 264857862 T^{3} + 655563 p^{3} T^{4} + 849 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 34 T + 66289 T^{2} - 150059212 T^{3} + 66289 p^{3} T^{4} - 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 201 T + 222207 T^{2} - 124662814 T^{3} + 222207 p^{3} T^{4} - 201 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 308 T + 801869 T^{2} + 190465256 T^{3} + 801869 p^{3} T^{4} + 308 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 66 T + 296733 T^{2} - 166151140 T^{3} + 296733 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1270 T + 1616266 T^{2} - 1036013462 T^{3} + 1616266 p^{3} T^{4} - 1270 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2947 T + 4307695 T^{2} - 3793644986 T^{3} + 4307695 p^{3} T^{4} - 2947 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1505 T + 1532325 T^{2} - 1057356194 T^{3} + 1532325 p^{3} T^{4} - 1505 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2153 T + 1729053 T^{2} - 1032545114 T^{3} + 1729053 p^{3} T^{4} - 2153 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1393 T - 492373 T^{2} - 1864511306 T^{3} - 492373 p^{3} T^{4} + 1393 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.434219196333803785221263983557, −9.103400405341792410842809595888, −8.627382617658954185419770509758, −8.509967965626324788990681416459, −7.76511661110027346142469405445, −7.63354838047929415638848573754, −7.62555634043328867365038216743, −7.13205415610739564695467706858, −6.69032204597692809376112540596, −6.54226184326504784463074311133, −6.10860662395109567376425385052, −6.08561080366823132856184852577, −5.85314685984784611965324026356, −5.04789733722208350574121270815, −4.78453587321693844273446064714, −4.77277880426358965602094556850, −3.58066894464677712556096605596, −3.58012563342075373400641806554, −3.43082849643593786744551792812, −2.22965344224799138776112739128, −2.07553221274814485542766158207, −1.93927826472050561523680970586, −0.963373725784560105299927875020, −0.63507882741980108753532686840, −0.13088512070240261168547361879,
0.13088512070240261168547361879, 0.63507882741980108753532686840, 0.963373725784560105299927875020, 1.93927826472050561523680970586, 2.07553221274814485542766158207, 2.22965344224799138776112739128, 3.43082849643593786744551792812, 3.58012563342075373400641806554, 3.58066894464677712556096605596, 4.77277880426358965602094556850, 4.78453587321693844273446064714, 5.04789733722208350574121270815, 5.85314685984784611965324026356, 6.08561080366823132856184852577, 6.10860662395109567376425385052, 6.54226184326504784463074311133, 6.69032204597692809376112540596, 7.13205415610739564695467706858, 7.62555634043328867365038216743, 7.63354838047929415638848573754, 7.76511661110027346142469405445, 8.509967965626324788990681416459, 8.627382617658954185419770509758, 9.103400405341792410842809595888, 9.434219196333803785221263983557
