| L(s) = 1 | − 2-s − 3·5-s − 4·7-s + 8-s + 3·10-s − 9·11-s + 6·13-s + 4·14-s − 3·16-s − 2·17-s − 4·19-s + 9·22-s − 23-s + 6·25-s − 6·26-s + 3·29-s + 2·34-s + 12·35-s + 13·37-s + 4·38-s − 3·40-s − 13·41-s − 13·43-s + 46-s − 2·47-s + 2·49-s − 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s − 1.51·7-s + 0.353·8-s + 0.948·10-s − 2.71·11-s + 1.66·13-s + 1.06·14-s − 3/4·16-s − 0.485·17-s − 0.917·19-s + 1.91·22-s − 0.208·23-s + 6/5·25-s − 1.17·26-s + 0.557·29-s + 0.342·34-s + 2.02·35-s + 2.13·37-s + 0.648·38-s − 0.474·40-s − 2.03·41-s − 1.98·43-s + 0.147·46-s − 0.291·47-s + 2/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 2 p T^{2} + 6 p T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} - 154 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 40 T^{2} + 60 T^{3} + 40 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 64 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} - 66 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 706 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1094 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 13 T + 131 T^{2} + 810 T^{3} + 131 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 72 T^{2} - 78 T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 45 T^{2} - 634 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 22 T + 317 T^{2} + 2852 T^{3} + 317 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 187 T^{2} - 1108 T^{3} + 187 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 28 T + 406 T^{2} + 3946 T^{3} + 406 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 488 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 35 T^{2} + 10 p T^{3} + 35 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 97 T^{2} + 92 T^{3} + 97 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 311 T^{2} - 2534 T^{3} + 311 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 450 T^{2} + 4738 T^{3} + 450 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + T + 149 T^{2} + 270 T^{3} + 149 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.016433213708273535051681906420, −8.528084431409831527464243263096, −8.366663494210071413922595888563, −8.255787825238577410852060673996, −7.973763008844740036931959743904, −7.82574119170216455372999927467, −7.33866108089632806735582357876, −7.14806405725519381589882957738, −6.86428818582604089364958964849, −6.46448378708401534638946222781, −6.27967064631483561479141928563, −6.03841958613424277644585438115, −5.75176145925009091942338120077, −5.14625513316087359525840431025, −4.89447070002354091948459157011, −4.83054435844237780996654853200, −4.28320081259642496990869590765, −3.93279128336953048778324803374, −3.73848039120127774903779037673, −3.26823824358493386232514481078, −2.93092831436610620795706668970, −2.67682380694961786233761682114, −2.45958085695007722759384711939, −1.50569963813351064387122547282, −1.34718162659656089720386186080, 0, 0, 0,
1.34718162659656089720386186080, 1.50569963813351064387122547282, 2.45958085695007722759384711939, 2.67682380694961786233761682114, 2.93092831436610620795706668970, 3.26823824358493386232514481078, 3.73848039120127774903779037673, 3.93279128336953048778324803374, 4.28320081259642496990869590765, 4.83054435844237780996654853200, 4.89447070002354091948459157011, 5.14625513316087359525840431025, 5.75176145925009091942338120077, 6.03841958613424277644585438115, 6.27967064631483561479141928563, 6.46448378708401534638946222781, 6.86428818582604089364958964849, 7.14806405725519381589882957738, 7.33866108089632806735582357876, 7.82574119170216455372999927467, 7.973763008844740036931959743904, 8.255787825238577410852060673996, 8.366663494210071413922595888563, 8.528084431409831527464243263096, 9.016433213708273535051681906420
