| L(s) = 1 | + 3·3-s − 2·5-s − 9·7-s + 6·9-s + 2·11-s + 4·13-s − 6·15-s + 3·17-s − 7·19-s − 27·21-s − 23-s − 6·25-s + 10·27-s − 11·29-s − 13·31-s + 6·33-s + 18·35-s − 5·37-s + 12·39-s − 41-s − 6·43-s − 12·45-s − 11·47-s + 37·49-s + 9·51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s − 3.40·7-s + 2·9-s + 0.603·11-s + 1.10·13-s − 1.54·15-s + 0.727·17-s − 1.60·19-s − 5.89·21-s − 0.208·23-s − 6/5·25-s + 1.92·27-s − 2.04·29-s − 2.33·31-s + 1.04·33-s + 3.04·35-s − 0.821·37-s + 1.92·39-s − 0.156·41-s − 0.914·43-s − 1.78·45-s − 1.60·47-s + 37/7·49-s + 1.26·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 18 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 9 T + 44 T^{2} + 142 T^{3} + 44 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 48 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 32 T^{2} - 6 p T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 8 T^{2} - 4 T^{3} + 8 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 7 T + 68 T^{2} + 270 T^{3} + 68 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 42 T^{2} - 18 T^{3} + 42 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 11 T + 96 T^{2} + 564 T^{3} + 96 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 13 T + 130 T^{2} + 778 T^{3} + 130 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 92 T^{2} + 384 T^{3} + 92 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 84 T^{2} + 156 T^{3} + 84 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 38 T^{2} - 76 T^{3} + 38 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 11 T + 104 T^{2} + 538 T^{3} + 104 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 70 T^{2} - 270 T^{3} + 70 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + T + 82 T^{2} + 220 T^{3} + 82 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 82 T^{2} + 556 T^{3} + 82 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 26 T + 406 T^{2} + 4116 T^{3} + 406 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 7 T + 78 T^{2} - 304 T^{3} + 78 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 206 T^{2} + 348 T^{3} + 206 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 3 T + 50 T^{2} - 350 T^{3} + 50 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 23 T + 358 T^{2} + 3816 T^{3} + 358 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 266 T^{2} - 2514 T^{3} + 266 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.340675897521777457356740296536, −7.79576474382844908263516865299, −7.64765110918979118682210731637, −7.50269952414268961370882379223, −7.25820508138248960844403154624, −6.86934254609784942382610088393, −6.66250887431661279031372286843, −6.36914753760477456525759336759, −6.28060759382446724887511886433, −6.15226307420011633173504766031, −5.47685947982184373706256412999, −5.37665928387075619821403573380, −5.28493609143822908908787828110, −4.24849519250029584635881358310, −4.18649628743495370330160109050, −4.07581203651196201461264081858, −3.66552095785634142075693185452, −3.59144151863374021969660379886, −3.44018732107085668248533532439, −2.92011767899068907082842729392, −2.91851417580428759900232288941, −2.49438483542100035042474985892, −1.83231473040240916365847419289, −1.55909844275077250496914044496, −1.46238698901718465497944506666, 0, 0, 0,
1.46238698901718465497944506666, 1.55909844275077250496914044496, 1.83231473040240916365847419289, 2.49438483542100035042474985892, 2.91851417580428759900232288941, 2.92011767899068907082842729392, 3.44018732107085668248533532439, 3.59144151863374021969660379886, 3.66552095785634142075693185452, 4.07581203651196201461264081858, 4.18649628743495370330160109050, 4.24849519250029584635881358310, 5.28493609143822908908787828110, 5.37665928387075619821403573380, 5.47685947982184373706256412999, 6.15226307420011633173504766031, 6.28060759382446724887511886433, 6.36914753760477456525759336759, 6.66250887431661279031372286843, 6.86934254609784942382610088393, 7.25820508138248960844403154624, 7.50269952414268961370882379223, 7.64765110918979118682210731637, 7.79576474382844908263516865299, 8.340675897521777457356740296536
