| L(s) = 1 | − 3·3-s − 3·5-s + 6·7-s + 6·9-s − 2·11-s + 9·15-s − 6·17-s + 2·19-s − 18·21-s − 3·23-s + 6·25-s − 10·27-s − 6·29-s + 6·31-s + 6·33-s − 18·35-s − 8·37-s − 14·41-s + 4·43-s − 18·45-s + 18·47-s + 10·49-s + 18·51-s − 18·53-s + 6·55-s − 6·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 2.26·7-s + 2·9-s − 0.603·11-s + 2.32·15-s − 1.45·17-s + 0.458·19-s − 3.92·21-s − 0.625·23-s + 6/5·25-s − 1.92·27-s − 1.11·29-s + 1.07·31-s + 1.04·33-s − 3.04·35-s − 1.31·37-s − 2.18·41-s + 0.609·43-s − 2.68·45-s + 2.62·47-s + 10/7·49-s + 2.52·51-s − 2.47·53-s + 0.809·55-s − 0.794·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 6 T + 26 T^{2} - 80 T^{3} + 26 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 12 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 76 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 118 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 68 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 56 T^{2} + 226 T^{3} + 56 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 308 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 76 T^{2} + 614 T^{3} + 76 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 14 T + 4 p T^{2} + 1090 T^{3} + 4 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} + 152 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 18 T + 183 T^{2} - 1420 T^{3} + 183 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 18 T + 228 T^{2} + 1906 T^{3} + 228 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 182 T^{2} + 1352 T^{3} + 182 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 261 T^{2} + 2068 T^{3} + 261 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 250 T^{2} - 1880 T^{3} + 250 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T - 22 T^{2} + 808 T^{3} - 22 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 81 T^{2} - 596 T^{3} + 81 p T^{4} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 154 T^{2} - 828 T^{3} + 154 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 283 T^{2} + 1764 T^{3} + 283 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 227 T^{2} - 516 T^{3} + 227 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.63430124942084123580263509458, −7.36532116527335794893382330537, −7.18547976291157517416223434673, −6.86498830181237395828405533205, −6.48341645184779360824986760616, −6.34933018486822462938110348275, −6.32218846260777555852782241807, −5.75522403732959192226188888901, −5.49918390735038014622692003198, −5.41283413927970937533299475125, −4.96586672763093458840562415882, −4.84451512974549818308643508356, −4.81117735760521487376947313780, −4.36833850010453642949169466723, −4.35662415081029902359188600206, −4.02198581463997334567437713243, −3.57344599659914293099909635860, −3.36838442365326328138031887682, −3.17768872231806716615204314090, −2.41242431322483931380709153845, −2.22704616181671861735379687336, −2.11503934858634596668419336340, −1.38476603141504015829808084943, −1.23935795496096494077652916711, −1.18653349009750679583664561685, 0, 0, 0,
1.18653349009750679583664561685, 1.23935795496096494077652916711, 1.38476603141504015829808084943, 2.11503934858634596668419336340, 2.22704616181671861735379687336, 2.41242431322483931380709153845, 3.17768872231806716615204314090, 3.36838442365326328138031887682, 3.57344599659914293099909635860, 4.02198581463997334567437713243, 4.35662415081029902359188600206, 4.36833850010453642949169466723, 4.81117735760521487376947313780, 4.84451512974549818308643508356, 4.96586672763093458840562415882, 5.41283413927970937533299475125, 5.49918390735038014622692003198, 5.75522403732959192226188888901, 6.32218846260777555852782241807, 6.34933018486822462938110348275, 6.48341645184779360824986760616, 6.86498830181237395828405533205, 7.18547976291157517416223434673, 7.36532116527335794893382330537, 7.63430124942084123580263509458
