| L(s) = 1 | − 4·2-s + 7·4-s + 4·7-s − 12·8-s − 33·11-s − 16·14-s − 7·16-s − 218·17-s + 146·19-s + 132·22-s − 200·23-s + 28·28-s − 68·29-s − 68·31-s + 28·32-s + 872·34-s + 390·37-s − 584·38-s + 196·41-s + 524·43-s − 231·44-s + 800·46-s − 60·47-s − 585·49-s − 158·53-s − 48·56-s + 272·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 7/8·4-s + 0.215·7-s − 0.530·8-s − 0.904·11-s − 0.305·14-s − 0.109·16-s − 3.11·17-s + 1.76·19-s + 1.27·22-s − 1.81·23-s + 0.188·28-s − 0.435·29-s − 0.393·31-s + 0.154·32-s + 4.39·34-s + 1.73·37-s − 2.49·38-s + 0.746·41-s + 1.85·43-s − 0.791·44-s + 2.56·46-s − 0.186·47-s − 1.70·49-s − 0.409·53-s − 0.114·56-s + 0.615·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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
−8.091805856604266478594351273548, −7.62092462870230603397638179250, −7.54310339024478392100580455064, −7.23038524149656420897761852591, −7.22355685569155139625425305056, −6.68287295455919677844783055464, −6.43578593991956318885773891325, −6.16646293377633518509964780034, −6.05909483957842277427810226753, −5.62435430876121234328333887383, −5.35079208258557067545549464792, −5.02419138052494839515410856903, −4.87914871996210602206837948778, −4.26488221513102203152465989462, −4.19873635862510114962608470971, −4.05722170900786590029058753528, −3.56710631193135148368292059804, −3.16895847201154798074544776221, −2.66639291894470737845918107640, −2.48982070466241759634048336774, −2.19846173862322036140236658924, −2.07528586890735883790936705105, −1.52283037107345849061114120788, −0.981325976330552932039369362914, −0.854566411509687942862874167235, 0, 0, 0,
0.854566411509687942862874167235, 0.981325976330552932039369362914, 1.52283037107345849061114120788, 2.07528586890735883790936705105, 2.19846173862322036140236658924, 2.48982070466241759634048336774, 2.66639291894470737845918107640, 3.16895847201154798074544776221, 3.56710631193135148368292059804, 4.05722170900786590029058753528, 4.19873635862510114962608470971, 4.26488221513102203152465989462, 4.87914871996210602206837948778, 5.02419138052494839515410856903, 5.35079208258557067545549464792, 5.62435430876121234328333887383, 6.05909483957842277427810226753, 6.16646293377633518509964780034, 6.43578593991956318885773891325, 6.68287295455919677844783055464, 7.22355685569155139625425305056, 7.23038524149656420897761852591, 7.54310339024478392100580455064, 7.62092462870230603397638179250, 8.091805856604266478594351273548
