| 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
−7.896780907587189343885172475733, −7.53565894674384363523676564802, −7.38929306430245737405458067310, −7.32791320621660964231821414561, −6.83139639748571341907716405261, −6.75442892845714426146830534631, −6.19769181025659281087586849714, −6.04269380845087318520484550118, −5.84250514622499080778865442389, −5.67212422220794682046227082179, −4.96573648879495973866344767020, −4.96140584927743854577816492776, −4.75373645046229564544091207678, −4.36168206337704547172371537328, −3.85435078937972752349167592293, −3.63440956760380372215173011894, −3.47484667335726850016520009413, −3.22621829996976447455509762593, −2.88966954708760825186489255770, −2.42767802305796288916999109395, −2.39334166901707793521730267709, −2.17436977427080183615326978743, −1.71350233085880468633582488525, −1.27112120808680496365448716562, −1.13735389206585621862625183552, 0, 0, 0,
1.13735389206585621862625183552, 1.27112120808680496365448716562, 1.71350233085880468633582488525, 2.17436977427080183615326978743, 2.39334166901707793521730267709, 2.42767802305796288916999109395, 2.88966954708760825186489255770, 3.22621829996976447455509762593, 3.47484667335726850016520009413, 3.63440956760380372215173011894, 3.85435078937972752349167592293, 4.36168206337704547172371537328, 4.75373645046229564544091207678, 4.96140584927743854577816492776, 4.96573648879495973866344767020, 5.67212422220794682046227082179, 5.84250514622499080778865442389, 6.04269380845087318520484550118, 6.19769181025659281087586849714, 6.75442892845714426146830534631, 6.83139639748571341907716405261, 7.32791320621660964231821414561, 7.38929306430245737405458067310, 7.53565894674384363523676564802, 7.896780907587189343885172475733
