| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 6·5-s + 9·6-s − 3·7-s + 10·8-s + 6·9-s − 18·10-s − 12·11-s + 18·12-s − 9·14-s − 18·15-s + 15·16-s − 12·17-s + 18·18-s − 36·20-s − 9·21-s − 36·22-s − 12·23-s + 30·24-s + 12·25-s + 10·27-s − 18·28-s − 6·29-s − 54·30-s − 3·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 2.68·5-s + 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s − 5.69·10-s − 3.61·11-s + 5.19·12-s − 2.40·14-s − 4.64·15-s + 15/4·16-s − 2.91·17-s + 4.24·18-s − 8.04·20-s − 1.96·21-s − 7.67·22-s − 2.50·23-s + 6.12·24-s + 12/5·25-s + 1.92·27-s − 3.40·28-s − 1.11·29-s − 9.85·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 63 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 43 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T + 78 T^{2} + 315 T^{3} + 78 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_6$ | \( 1 + 19 T^{3} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 27 p T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 549 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 297 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 169 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 102 T^{2} - 203 T^{3} + 102 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 3 p T^{2} + 711 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 259 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1251 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 3 T + 78 T^{2} + 99 T^{3} + 78 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 60 T^{2} + 153 T^{3} + 60 p T^{4} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 159 T^{2} + 367 T^{3} + 159 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 174 T^{2} + 715 T^{3} + 174 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 3 p T^{2} + 1197 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T - 15 T^{2} + 301 T^{3} - 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 189 T^{2} - 1349 T^{3} + 189 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 3 T - 21 T^{2} + 1395 T^{3} - 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T - 3 T^{2} + 1359 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1051 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.325429143533674064325373990886, −7.933817003779015190342910496849, −7.890991864103708859556483320694, −7.81974885616524817472191612414, −7.30461466378993975813350729831, −7.27565899611669356034811712442, −7.02728765722013042853642636203, −6.41648168573459790601606815175, −6.40381390401041954300857703730, −6.21422121927894644665592941851, −5.66985031241079195999153705539, −5.26230852022658404317002565540, −5.05947302160295411880069060114, −4.80814198104560662311475307017, −4.42653381591519166930568647537, −4.27641407707610209618361783596, −3.98129765993333304517105581890, −3.69036623563143435470232950365, −3.56756897708293649236956001246, −3.02737662631799433248367248412, −2.95701712481061046298557494549, −2.81164817411683596592111213932, −2.06718988207253053120553986752, −1.99830278932572404134926175382, −1.90862217838124315608554626132, 0, 0, 0,
1.90862217838124315608554626132, 1.99830278932572404134926175382, 2.06718988207253053120553986752, 2.81164817411683596592111213932, 2.95701712481061046298557494549, 3.02737662631799433248367248412, 3.56756897708293649236956001246, 3.69036623563143435470232950365, 3.98129765993333304517105581890, 4.27641407707610209618361783596, 4.42653381591519166930568647537, 4.80814198104560662311475307017, 5.05947302160295411880069060114, 5.26230852022658404317002565540, 5.66985031241079195999153705539, 6.21422121927894644665592941851, 6.40381390401041954300857703730, 6.41648168573459790601606815175, 7.02728765722013042853642636203, 7.27565899611669356034811712442, 7.30461466378993975813350729831, 7.81974885616524817472191612414, 7.890991864103708859556483320694, 7.933817003779015190342910496849, 8.325429143533674064325373990886
