| L(s) = 1 | − 3·3-s + 5-s + 3·7-s + 6·9-s + 4·11-s − 5·13-s − 3·15-s − 4·17-s + 2·19-s − 9·21-s + 3·23-s − 9·25-s − 10·27-s − 2·29-s − 2·31-s − 12·33-s + 3·35-s − 14·37-s + 15·39-s − 16·41-s + 9·43-s + 6·45-s − 10·47-s + 6·49-s + 12·51-s + 53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 1.13·7-s + 2·9-s + 1.20·11-s − 1.38·13-s − 0.774·15-s − 0.970·17-s + 0.458·19-s − 1.96·21-s + 0.625·23-s − 9/5·25-s − 1.92·27-s − 0.371·29-s − 0.359·31-s − 2.08·33-s + 0.507·35-s − 2.30·37-s + 2.40·39-s − 2.49·41-s + 1.37·43-s + 0.894·45-s − 1.45·47-s + 6/7·49-s + 1.68·51-s + 0.137·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 12 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 32 T^{2} - 80 T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 28 T^{2} + 128 T^{3} + 28 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 44 T^{2} + 110 T^{3} + 44 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 26 T^{2} - 108 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 82 T^{2} + 114 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 156 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 170 T^{2} + 30 p T^{3} + 170 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 176 T^{2} + 1238 T^{3} + 176 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 152 T^{2} - 790 T^{3} + 152 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 17 T^{2} - 244 T^{3} + 17 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T - 20 T^{2} - 20 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 17 T + 268 T^{2} - 2154 T^{3} + 268 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 182 T^{2} + 364 T^{3} + 182 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 90 T^{2} + 478 T^{3} + 90 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - T + 56 T^{2} + 274 T^{3} + 56 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 38 T + 694 T^{2} + 7502 T^{3} + 694 p T^{4} + 38 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 10 T + 258 T^{2} - 1564 T^{3} + 258 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 180 T^{2} + 1224 T^{3} + 180 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + T + 116 T^{2} + 836 T^{3} + 116 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 274 T^{2} + 1678 T^{3} + 274 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.26552733976399072432180783877, −7.13375951367453045568039632272, −6.91628907930372963836521079968, −6.55359542014422620599201444227, −6.36353945236985926590843048538, −6.10599793801371604136394843859, −5.74717356347840943021861397705, −5.68139952668086965078713808687, −5.43049236875561190862049851346, −5.27137477934515055466974454360, −4.90296606312278505527734519319, −4.74510633141001600890146618113, −4.64178906631413710725656907329, −4.17031416480320096000324276464, −4.02230325087071911645390794700, −3.96435730573117127229746463035, −3.28885341994766267067931154384, −3.23732779323222932598160179286, −2.92727112193938768399828031311, −2.11764403899645822654593012790, −2.11439243022250460751533825598, −2.04881613349570059573026247163, −1.46188195437792208920578892405, −1.21079396516726511565422059844, −1.16066472345007340551399165282, 0, 0, 0,
1.16066472345007340551399165282, 1.21079396516726511565422059844, 1.46188195437792208920578892405, 2.04881613349570059573026247163, 2.11439243022250460751533825598, 2.11764403899645822654593012790, 2.92727112193938768399828031311, 3.23732779323222932598160179286, 3.28885341994766267067931154384, 3.96435730573117127229746463035, 4.02230325087071911645390794700, 4.17031416480320096000324276464, 4.64178906631413710725656907329, 4.74510633141001600890146618113, 4.90296606312278505527734519319, 5.27137477934515055466974454360, 5.43049236875561190862049851346, 5.68139952668086965078713808687, 5.74717356347840943021861397705, 6.10599793801371604136394843859, 6.36353945236985926590843048538, 6.55359542014422620599201444227, 6.91628907930372963836521079968, 7.13375951367453045568039632272, 7.26552733976399072432180783877
