| L(s) = 1 | + 3·2-s − 2·3-s + 6·4-s + 2·5-s − 6·6-s + 4·7-s + 10·8-s − 3·9-s + 6·10-s − 3·11-s − 12·12-s − 7·13-s + 12·14-s − 4·15-s + 15·16-s + 16·17-s − 9·18-s + 12·20-s − 8·21-s − 9·22-s + 8·23-s − 20·24-s + 25-s − 21·26-s + 10·27-s + 24·28-s − 5·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 3·4-s + 0.894·5-s − 2.44·6-s + 1.51·7-s + 3.53·8-s − 9-s + 1.89·10-s − 0.904·11-s − 3.46·12-s − 1.94·13-s + 3.20·14-s − 1.03·15-s + 15/4·16-s + 3.88·17-s − 2.12·18-s + 2.68·20-s − 1.74·21-s − 1.91·22-s + 1.66·23-s − 4.08·24-s + 1/5·25-s − 4.11·26-s + 1.92·27-s + 4.53·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.53006804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.53006804\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 18 T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 7 T + 46 T^{2} + 163 T^{3} + 46 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 16 T + 123 T^{2} - 606 T^{3} + 123 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 29 T^{2} - 64 T^{3} + 29 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 74 T^{2} + 265 T^{3} + 74 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 376 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 91 T^{2} + 98 T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 119 T^{2} - 2 T^{3} + 119 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - T + 116 T^{2} - 63 T^{3} + 116 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 134 T^{2} + 833 T^{3} + 134 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 151 T^{2} + 408 T^{3} + 151 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 161 T^{2} - 16 T^{3} + 161 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 21 T + 282 T^{2} - 2549 T^{3} + 282 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 191 T^{2} + 1602 T^{3} + 191 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 116 T^{2} + 577 T^{3} + 116 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 26 T + 395 T^{2} + 3918 T^{3} + 395 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 199 T^{2} - 1502 T^{3} + 199 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 7 T + 172 T^{2} - 1293 T^{3} + 172 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 23 T + 350 T^{2} + 3691 T^{3} + 350 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 13 T - 26 T^{2} + 1703 T^{3} - 26 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.00385223482533950815954317912, −6.30182259196858358297019955636, −6.27547135666425119799987197612, −6.23588118439926375491023688273, −5.64854433364585405013648252659, −5.56832804308581033084264941512, −5.50311294130374849795742389044, −5.33946149228023746254689032169, −5.08170472716196673095723019495, −5.03193868782788365118838474115, −4.66096391898261609597404335477, −4.53519395118940788334129010470, −4.29978492047355749903150578152, −3.71750958390914318509906962988, −3.42582760140596882199767871819, −3.23486835926649009876596567327, −2.97156895743206457809874320326, −2.84948313688816400468910559563, −2.60401089446491936910047286477, −2.05015631658730302663915446146, −1.99737661115646137639445331963, −1.50090630076029300836934688342, −1.31431382228710061291835523231, −0.68700383959030718329347893577, −0.54555351085154664883248145984,
0.54555351085154664883248145984, 0.68700383959030718329347893577, 1.31431382228710061291835523231, 1.50090630076029300836934688342, 1.99737661115646137639445331963, 2.05015631658730302663915446146, 2.60401089446491936910047286477, 2.84948313688816400468910559563, 2.97156895743206457809874320326, 3.23486835926649009876596567327, 3.42582760140596882199767871819, 3.71750958390914318509906962988, 4.29978492047355749903150578152, 4.53519395118940788334129010470, 4.66096391898261609597404335477, 5.03193868782788365118838474115, 5.08170472716196673095723019495, 5.33946149228023746254689032169, 5.50311294130374849795742389044, 5.56832804308581033084264941512, 5.64854433364585405013648252659, 6.23588118439926375491023688273, 6.27547135666425119799987197612, 6.30182259196858358297019955636, 7.00385223482533950815954317912
