| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 3·5-s − 9·6-s + 3·7-s + 10·8-s − 9·10-s − 3·11-s − 18·12-s + 9·14-s + 9·15-s + 15·16-s − 18·20-s − 9·21-s − 9·22-s − 6·23-s − 30·24-s − 6·25-s + 12·27-s + 18·28-s − 18·29-s + 27·30-s + 3·31-s + 21·32-s + 9·33-s − 9·35-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s − 2.84·10-s − 0.904·11-s − 5.19·12-s + 2.40·14-s + 2.32·15-s + 15/4·16-s − 4.02·20-s − 1.96·21-s − 1.91·22-s − 1.25·23-s − 6.12·24-s − 6/5·25-s + 2.30·27-s + 3.40·28-s − 3.34·29-s + 4.92·30-s + 0.538·31-s + 3.71·32-s + 1.56·33-s − 1.52·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + p T + p^{2} T^{2} + 5 p T^{3} + p^{3} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 29 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 65 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 36 T^{2} - T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 24 T^{2} - 27 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 284 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 18 T + 186 T^{2} + 1215 T^{3} + 186 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 133 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 9 T + 81 T^{2} - 359 T^{3} + 81 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 57 T^{2} - 505 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 15 T + 168 T^{2} + 1163 T^{3} + 168 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 159 T^{2} - 837 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1199 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 155 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 18 T + 279 T^{2} + 2332 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 3 T + 57 T^{2} + 87 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 21 T + 327 T^{2} + 3047 T^{3} + 327 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 6 T + 174 T^{2} + 931 T^{3} + 174 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 15 T + 297 T^{2} - 2507 T^{3} + 297 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 15 T + 114 T^{2} + 351 T^{3} + 114 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 2136 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.70353605280294247213062073855, −7.27885442784541754962586352949, −7.24211030308274160518927988843, −6.87312884115100455006396356450, −6.31385513891749868783019038365, −6.15887238697552127042591396929, −6.10754534088389438249603745371, −5.84721320615097025892865316409, −5.70416462077090159438226423849, −5.47208861161116404306707135551, −5.12010697129637766479716975541, −4.96048759842280866786746061566, −4.83128128282586143793377161622, −4.34046797664777313266814298624, −4.27394665307009153602069525368, −4.03050461308935306128756949944, −3.68487258486702079381383305646, −3.38697651352457643499061924421, −3.35475878394219619750942694072, −2.70425115201818181539622735213, −2.37595654697367703214665232638, −2.33506187518766204656078071439, −1.88101220165856087921501759000, −1.29897141774553520352100295367, −1.22153408448666841029150092843, 0, 0, 0,
1.22153408448666841029150092843, 1.29897141774553520352100295367, 1.88101220165856087921501759000, 2.33506187518766204656078071439, 2.37595654697367703214665232638, 2.70425115201818181539622735213, 3.35475878394219619750942694072, 3.38697651352457643499061924421, 3.68487258486702079381383305646, 4.03050461308935306128756949944, 4.27394665307009153602069525368, 4.34046797664777313266814298624, 4.83128128282586143793377161622, 4.96048759842280866786746061566, 5.12010697129637766479716975541, 5.47208861161116404306707135551, 5.70416462077090159438226423849, 5.84721320615097025892865316409, 6.10754534088389438249603745371, 6.15887238697552127042591396929, 6.31385513891749868783019038365, 6.87312884115100455006396356450, 7.24211030308274160518927988843, 7.27885442784541754962586352949, 7.70353605280294247213062073855
