| L(s) = 1 | + 4·2-s + 9·3-s + 7·4-s + 36·6-s + 4·7-s + 12·8-s + 54·9-s + 33·11-s + 63·12-s + 16·14-s − 7·16-s + 218·17-s + 216·18-s + 146·19-s + 36·21-s + 132·22-s + 200·23-s + 108·24-s + 270·27-s + 28·28-s + 68·29-s − 68·31-s − 28·32-s + 297·33-s + 872·34-s + 378·36-s + 390·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 7/8·4-s + 2.44·6-s + 0.215·7-s + 0.530·8-s + 2·9-s + 0.904·11-s + 1.51·12-s + 0.305·14-s − 0.109·16-s + 3.11·17-s + 2.82·18-s + 1.76·19-s + 0.374·21-s + 1.27·22-s + 1.81·23-s + 0.918·24-s + 1.92·27-s + 0.188·28-s + 0.435·29-s − 0.393·31-s − 0.154·32-s + 1.56·33-s + 4.39·34-s + 7/4·36-s + 1.73·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(45.34513682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(45.34513682\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 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
−8.824538265957090862984442657588, −8.274669889890678787790145893236, −8.141253089791682904017279604648, −7.70231136854552248755973304148, −7.60649173355699630157614715211, −7.21641676367675934544609925841, −7.19538017647700608216589101276, −6.74517118727960106577098322585, −6.14842024987554433132806232754, −6.08708190831412417499210704726, −5.48565476795241888890576898314, −5.45186028111385934401011142552, −4.95279547076424105681938167816, −4.67588187526523506948114370755, −4.31951304257228794079168632437, −4.12390094827828533918595865420, −3.51096882456687860401636107810, −3.35697234964986496513408637015, −3.17039257360102443364282472464, −2.67814348647968432391518070514, −2.66408071374131149618080691913, −1.65470999538928771380172920643, −1.50455861492090021604115178052, −0.978278863960150269743198770832, −0.812636248968871468148556731499,
0.812636248968871468148556731499, 0.978278863960150269743198770832, 1.50455861492090021604115178052, 1.65470999538928771380172920643, 2.66408071374131149618080691913, 2.67814348647968432391518070514, 3.17039257360102443364282472464, 3.35697234964986496513408637015, 3.51096882456687860401636107810, 4.12390094827828533918595865420, 4.31951304257228794079168632437, 4.67588187526523506948114370755, 4.95279547076424105681938167816, 5.45186028111385934401011142552, 5.48565476795241888890576898314, 6.08708190831412417499210704726, 6.14842024987554433132806232754, 6.74517118727960106577098322585, 7.19538017647700608216589101276, 7.21641676367675934544609925841, 7.60649173355699630157614715211, 7.70231136854552248755973304148, 8.141253089791682904017279604648, 8.274669889890678787790145893236, 8.824538265957090862984442657588
