| L(s) = 1 | + 3·3-s − 3·5-s − 8·7-s + 6·9-s − 2·11-s + 4·13-s − 9·15-s + 4·17-s + 4·19-s − 24·21-s − 6·23-s + 6·25-s + 10·27-s − 2·29-s − 3·31-s − 6·33-s + 24·35-s + 10·37-s + 12·39-s − 8·41-s − 10·43-s − 18·45-s + 10·47-s + 27·49-s + 12·51-s + 6·55-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s − 3.02·7-s + 2·9-s − 0.603·11-s + 1.10·13-s − 2.32·15-s + 0.970·17-s + 0.917·19-s − 5.23·21-s − 1.25·23-s + 6/5·25-s + 1.92·27-s − 0.371·29-s − 0.538·31-s − 1.04·33-s + 4.05·35-s + 1.64·37-s + 1.92·39-s − 1.24·41-s − 1.52·43-s − 2.68·45-s + 1.45·47-s + 27/7·49-s + 1.68·51-s + 0.809·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 8 T + 37 T^{2} + 118 T^{3} + 37 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 20 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 98 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 132 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 25 T^{2} - 56 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 57 T^{2} + 272 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 21 T^{2} + 230 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 10 T + 139 T^{2} - 758 T^{3} + 139 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 608 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 141 T^{2} + 836 T^{3} + 141 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 89 T^{2} - 432 T^{3} + 89 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 135 T^{2} - 36 T^{3} + 135 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 187 T^{2} - 938 T^{3} + 187 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 + 18 T + 273 T^{2} + 2330 T^{3} + 273 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 147 T^{2} + 1162 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 279 T^{2} - 2522 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 333 T^{2} + 3124 T^{3} + 333 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 22 T + 325 T^{2} + 3160 T^{3} + 325 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 5 T^{2} - 30 T^{3} + 5 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 123 T^{2} - 64 T^{3} + 123 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.40778753109173572012781677151, −7.15418654600094475815254049824, −6.91142475646651734587028612605, −6.80677301236693434501980084216, −6.34604810361562021801944905498, −6.26974359568956614732909468392, −6.07102305949354961583652814087, −5.65831262539133229581415519384, −5.44685960345031303759820765073, −5.44308686489062930668853587345, −4.69228577866158452637453018737, −4.59582784102998162182901671335, −4.34465530979254364005947437384, −3.86069795207944480913786002575, −3.73267037835884610647556114971, −3.73031134491756510545134505540, −3.22627980601344521508547829326, −3.20666343524690106615393711186, −3.18867414274944541852562156086, −2.51038764187387803437759237763, −2.43645150179500340624857754306, −2.39418084514464639926931843532, −1.32303627365273485511921691362, −1.27826709303294391363430137420, −1.21655947750892312251002142585, 0, 0, 0,
1.21655947750892312251002142585, 1.27826709303294391363430137420, 1.32303627365273485511921691362, 2.39418084514464639926931843532, 2.43645150179500340624857754306, 2.51038764187387803437759237763, 3.18867414274944541852562156086, 3.20666343524690106615393711186, 3.22627980601344521508547829326, 3.73031134491756510545134505540, 3.73267037835884610647556114971, 3.86069795207944480913786002575, 4.34465530979254364005947437384, 4.59582784102998162182901671335, 4.69228577866158452637453018737, 5.44308686489062930668853587345, 5.44685960345031303759820765073, 5.65831262539133229581415519384, 6.07102305949354961583652814087, 6.26974359568956614732909468392, 6.34604810361562021801944905498, 6.80677301236693434501980084216, 6.91142475646651734587028612605, 7.15418654600094475815254049824, 7.40778753109173572012781677151
