| L(s) = 1 | − 3·2-s + 6·4-s + 5·5-s − 10·8-s − 15·10-s − 11-s − 2·13-s + 15·16-s + 4·17-s − 3·19-s + 30·20-s + 3·22-s − 7·23-s + 6·25-s + 6·26-s − 5·29-s − 14·31-s − 21·32-s − 12·34-s + 9·37-s + 9·38-s − 50·40-s + 12·41-s − 18·43-s − 6·44-s + 21·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 2.23·5-s − 3.53·8-s − 4.74·10-s − 0.301·11-s − 0.554·13-s + 15/4·16-s + 0.970·17-s − 0.688·19-s + 6.70·20-s + 0.639·22-s − 1.45·23-s + 6/5·25-s + 1.17·26-s − 0.928·29-s − 2.51·31-s − 3.71·32-s − 2.05·34-s + 1.47·37-s + 1.45·38-s − 7.90·40-s + 1.87·41-s − 2.74·43-s − 0.904·44-s + 3.09·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 - p T + 19 T^{2} - 47 T^{3} + 19 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 7 T^{2} + 5 p T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 36 T^{2} + 49 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 7 T^{2} + 32 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 107 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 73 T^{2} + 319 T^{3} + 73 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 55 T^{2} + 323 T^{3} + 55 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 138 T^{2} + 841 T^{3} + 138 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 102 T^{2} - 593 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 162 T^{2} - 1011 T^{3} + 162 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 210 T^{2} + 1549 T^{3} + 210 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 117 T^{2} + 309 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 963 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 76 T^{2} + 11 p T^{3} + 76 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 48 T^{2} + 229 T^{3} + 48 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 143 T^{2} + 521 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 7 T + 163 T^{2} + 895 T^{3} + 163 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 25 T + 371 T^{2} + 3601 T^{3} + 371 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 93 T^{2} + 335 T^{3} + 93 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 244 T^{2} + 1235 T^{3} + 244 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 261 T^{2} - 1539 T^{3} + 261 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 28 T + 527 T^{2} + 5968 T^{3} + 527 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.47455914085131843675533146072, −7.06850581447763956758293987959, −6.97668209438154203827545933997, −6.66414478368763677712766328637, −6.30579572397566991721128840178, −5.99711166261398426677534643086, −5.97350394427860832204758288593, −5.83825842625079001419116498163, −5.58761506743550376660138262727, −5.48443803771563606711393620861, −5.03543976100124350616603631297, −4.80375760202406052407185214004, −4.48463156557972897814131886907, −3.91077886064879235016995870676, −3.87415436798688027643126410097, −3.67462426021388066465960820244, −3.10056519593738239315752575820, −2.86931581109815888904306214955, −2.62085860071902581664451044346, −2.18889476993159030779857978527, −2.15546072295428746472679898339, −1.96843984235262030505773212436, −1.43153216856913565649301385660, −1.26168539677479503495145849879, −1.24305253637217689552194798826, 0, 0, 0,
1.24305253637217689552194798826, 1.26168539677479503495145849879, 1.43153216856913565649301385660, 1.96843984235262030505773212436, 2.15546072295428746472679898339, 2.18889476993159030779857978527, 2.62085860071902581664451044346, 2.86931581109815888904306214955, 3.10056519593738239315752575820, 3.67462426021388066465960820244, 3.87415436798688027643126410097, 3.91077886064879235016995870676, 4.48463156557972897814131886907, 4.80375760202406052407185214004, 5.03543976100124350616603631297, 5.48443803771563606711393620861, 5.58761506743550376660138262727, 5.83825842625079001419116498163, 5.97350394427860832204758288593, 5.99711166261398426677534643086, 6.30579572397566991721128840178, 6.66414478368763677712766328637, 6.97668209438154203827545933997, 7.06850581447763956758293987959, 7.47455914085131843675533146072
