| L(s) = 1 | − 3·2-s − 2·3-s + 6·4-s − 3·5-s + 6·6-s − 6·7-s − 10·8-s + 9-s + 9·10-s + 6·11-s − 12·12-s − 4·13-s + 18·14-s + 6·15-s + 15·16-s + 8·17-s − 3·18-s + 2·19-s − 18·20-s + 12·21-s − 18·22-s + 14·23-s + 20·24-s + 6·25-s + 12·26-s − 36·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3·4-s − 1.34·5-s + 2.44·6-s − 2.26·7-s − 3.53·8-s + 1/3·9-s + 2.84·10-s + 1.80·11-s − 3.46·12-s − 1.10·13-s + 4.81·14-s + 1.54·15-s + 15/4·16-s + 1.94·17-s − 0.707·18-s + 0.458·19-s − 4.02·20-s + 2.61·21-s − 3.83·22-s + 2.91·23-s + 4.08·24-s + 6/5·25-s + 2.35·26-s − 6.80·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79507000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79507000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2698064110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2698064110\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 43 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 6 T + 26 T^{2} + 76 T^{3} + 26 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} - 2 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 57 T^{2} - 228 T^{3} + 57 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 42 T^{2} - 44 T^{3} + 42 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 119 T^{2} - 652 T^{3} + 119 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T - 4 T^{2} + 194 T^{3} - 4 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 54 T^{2} - 16 T^{3} + 54 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 300 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 148 T^{2} - 978 T^{3} + 148 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 135 T^{2} + 180 T^{3} + 135 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 22 T + 223 T^{2} - 1644 T^{3} + 223 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 16 T + 201 T^{2} - 1536 T^{3} + 201 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 84 T^{2} - 118 T^{3} + 84 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 24 T + 350 T^{2} + 3340 T^{3} + 350 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 191 T^{2} + 868 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 68 T^{2} - 42 T^{3} + 68 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 202 T^{2} + 640 T^{3} + 202 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 157 T^{2} - 168 T^{3} + 157 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 16 T + 121 T^{2} + 444 T^{3} + 121 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 233 T^{2} + 1260 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.918003861114741607136872846797, −9.820665683799804300756520580307, −9.271358355941244493737534813177, −9.118321377962181261660554227025, −8.853824093388744208139121635249, −8.570252140706039296530331760444, −8.307698417649849304584118619671, −7.51272775311896320618846045716, −7.32826815609867024189997284667, −7.21700257573961770093316931977, −7.11686656238056242629556476569, −6.63571810994769751882699899084, −6.27163115132777004663126909562, −6.07479910196577258716190728672, −5.46298706516002827987477761545, −5.42704304791645152829253897350, −4.71461887009654059920626984862, −4.05275533165323625217284410591, −3.86231430787676711132163894953, −3.07345969918674156346171202468, −2.96390400231112389804841259598, −2.83103159090140609664959358801, −1.47661923668323543711792828119, −0.851791562144416875283763427443, −0.57853157686639717974893972065,
0.57853157686639717974893972065, 0.851791562144416875283763427443, 1.47661923668323543711792828119, 2.83103159090140609664959358801, 2.96390400231112389804841259598, 3.07345969918674156346171202468, 3.86231430787676711132163894953, 4.05275533165323625217284410591, 4.71461887009654059920626984862, 5.42704304791645152829253897350, 5.46298706516002827987477761545, 6.07479910196577258716190728672, 6.27163115132777004663126909562, 6.63571810994769751882699899084, 7.11686656238056242629556476569, 7.21700257573961770093316931977, 7.32826815609867024189997284667, 7.51272775311896320618846045716, 8.307698417649849304584118619671, 8.570252140706039296530331760444, 8.853824093388744208139121635249, 9.118321377962181261660554227025, 9.271358355941244493737534813177, 9.820665683799804300756520580307, 9.918003861114741607136872846797
