| L(s) = 1 | + 4·2-s + 7·4-s + 15·5-s − 4·7-s + 12·8-s + 60·10-s − 33·11-s − 16·14-s − 7·16-s + 218·17-s + 146·19-s + 105·20-s − 132·22-s + 200·23-s + 150·25-s − 28·28-s − 68·29-s − 68·31-s − 28·32-s + 872·34-s − 60·35-s − 390·37-s + 584·38-s + 180·40-s + 196·41-s − 524·43-s − 231·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 7/8·4-s + 1.34·5-s − 0.215·7-s + 0.530·8-s + 1.89·10-s − 0.904·11-s − 0.305·14-s − 0.109·16-s + 3.11·17-s + 1.76·19-s + 1.17·20-s − 1.27·22-s + 1.81·23-s + 6/5·25-s − 0.188·28-s − 0.435·29-s − 0.393·31-s − 0.154·32-s + 4.39·34-s − 0.289·35-s − 1.73·37-s + 2.49·38-s + 0.711·40-s + 0.746·41-s − 1.85·43-s − 0.791·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(15.76662610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.76662610\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - p^{2} T + 9 T^{2} - 5 p^{2} T^{3} + 9 p^{3} T^{4} - p^{8} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 601 T^{2} - 1000 T^{3} + 601 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3031 T^{2} - 34144 T^{3} + 3031 p^{3} T^{4} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 218 T + 24419 T^{2} - 1906964 T^{3} + 24419 p^{3} T^{4} - 218 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 146 T + 25953 T^{2} - 2033788 T^{3} + 25953 p^{3} T^{4} - 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 200 T + 33829 T^{2} - 4868464 T^{3} + 33829 p^{3} T^{4} - 200 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 68 T + 18803 T^{2} + 153848 T^{3} + 18803 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 68 T + 66141 T^{2} + 2239480 T^{3} + 66141 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 390 T + 188419 T^{2} + 40128292 T^{3} + 188419 p^{3} T^{4} + 390 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 196 T + 4407 p T^{2} - 22652824 T^{3} + 4407 p^{4} T^{4} - 196 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 524 T + 209853 T^{2} + 52049416 T^{3} + 209853 p^{3} T^{4} + 524 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 60 T + 175549 T^{2} + 8508216 T^{3} + 175549 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 158 T + 185779 T^{2} - 86620084 T^{3} + 185779 p^{3} T^{4} - 158 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1044 T + 680281 T^{2} - 344604088 T^{3} + 680281 p^{3} T^{4} - 1044 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 642 T + 620395 T^{2} - 268686220 T^{3} + 620395 p^{3} T^{4} - 642 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 236 T + 440081 T^{2} + 229497800 T^{3} + 440081 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 544 T + 944005 T^{2} - 395960768 T^{3} + 944005 p^{3} T^{4} - 544 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 900 T + 867475 T^{2} - 705839944 T^{3} + 867475 p^{3} T^{4} - 900 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1586 T + 2041325 T^{2} + 1549224716 T^{3} + 2041325 p^{3} T^{4} + 1586 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1582 T + 1407717 T^{2} - 884488684 T^{3} + 1407717 p^{3} T^{4} - 1582 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2122 T + 3521847 T^{2} - 3285333068 T^{3} + 3521847 p^{3} T^{4} - 2122 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 618 T + 1446319 T^{2} - 1351607564 T^{3} + 1446319 p^{3} T^{4} - 618 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.645147905794819947867836996386, −9.064691858331732914345297756589, −8.705866672315565271473547443776, −8.577905468421247202418826271988, −7.973223279039062694412929075135, −7.71264587291252092193863394305, −7.34947819434773658278079070096, −7.25618065771403870770627620592, −6.64908105533986032076649167788, −6.55461965530677026412684506762, −6.02273943533626352805314420038, −5.63960419299917378855309782888, −5.37206852575750273361412474343, −5.07380067114678322166106845756, −4.99446325746117455415662326800, −4.94399065267532515937127494808, −3.84278545197312442400652300407, −3.60843608193586126963928075138, −3.39950478248093759197161883296, −2.97787144440632533449869752915, −2.62452493929847547921863494152, −2.07243390623481774410648642616, −1.51168657565424928911376892449, −1.06028125014971309244569921158, −0.62731432633879161527461794296,
0.62731432633879161527461794296, 1.06028125014971309244569921158, 1.51168657565424928911376892449, 2.07243390623481774410648642616, 2.62452493929847547921863494152, 2.97787144440632533449869752915, 3.39950478248093759197161883296, 3.60843608193586126963928075138, 3.84278545197312442400652300407, 4.94399065267532515937127494808, 4.99446325746117455415662326800, 5.07380067114678322166106845756, 5.37206852575750273361412474343, 5.63960419299917378855309782888, 6.02273943533626352805314420038, 6.55461965530677026412684506762, 6.64908105533986032076649167788, 7.25618065771403870770627620592, 7.34947819434773658278079070096, 7.71264587291252092193863394305, 7.973223279039062694412929075135, 8.577905468421247202418826271988, 8.705866672315565271473547443776, 9.064691858331732914345297756589, 9.645147905794819947867836996386
