L(s) = 1 | − 3·3-s + 3·4-s + 12·5-s + 3·7-s − 2·8-s + 3·9-s − 6·11-s − 9·12-s − 36·15-s + 6·16-s − 6·17-s − 9·19-s + 36·20-s − 9·21-s − 3·23-s + 6·24-s + 60·25-s + 2·27-s + 9·28-s − 9·29-s + 18·31-s − 9·32-s + 18·33-s + 36·35-s + 9·36-s − 12·37-s − 24·40-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 3/2·4-s + 5.36·5-s + 1.13·7-s − 0.707·8-s + 9-s − 1.80·11-s − 2.59·12-s − 9.29·15-s + 3/2·16-s − 1.45·17-s − 2.06·19-s + 8.04·20-s − 1.96·21-s − 0.625·23-s + 1.22·24-s + 12·25-s + 0.384·27-s + 1.70·28-s − 1.67·29-s + 3.23·31-s − 1.59·32-s + 3.13·33-s + 6.08·35-s + 3/2·36-s − 1.97·37-s − 3.79·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.701564215\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.701564215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + T + T^{2} )^{3} \) |
| 7 | \( ( 1 - T + T^{2} )^{3} \) |
| 13 | \( 1 - 19 T^{3} + p^{3} T^{6} \) |
good | 2 | \( 1 - 3 T^{2} + p T^{3} + 3 T^{4} - 3 T^{5} - T^{6} - 3 p T^{7} + 3 p^{2} T^{8} + p^{4} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( ( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 + 6 T + 12 T^{2} + 2 p T^{3} - 54 T^{4} - 978 T^{5} - 4309 T^{6} - 978 p T^{7} - 54 p^{2} T^{8} + 2 p^{4} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 6 T + 12 T^{2} + 54 T^{3} - 6 p T^{4} - 2082 T^{5} - 8345 T^{6} - 2082 p T^{7} - 6 p^{3} T^{8} + 54 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 9 T + 9 T^{2} - 2 T^{3} + 891 T^{4} + 2025 T^{5} - 8394 T^{6} + 2025 p T^{7} + 891 p^{2} T^{8} - 2 p^{3} T^{9} + 9 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 3 T - 24 T^{2} - 63 T^{3} + 87 T^{4} - 294 T^{5} + 1087 T^{6} - 294 p T^{7} + 87 p^{2} T^{8} - 63 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 9 T + 42 T^{2} - 47 T^{3} - 1401 T^{4} - 5028 T^{5} - 23851 T^{6} - 5028 p T^{7} - 1401 p^{2} T^{8} - 47 p^{3} T^{9} + 42 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 9 T + 3 p T^{2} - 531 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 12 T + 42 T^{2} + 102 T^{3} - 102 T^{4} - 16926 T^{5} - 166057 T^{6} - 16926 p T^{7} - 102 p^{2} T^{8} + 102 p^{3} T^{9} + 42 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 + 6 T + 45 T^{2} + 674 T^{3} + 2634 T^{4} + 17790 T^{5} + 206501 T^{6} + 17790 p T^{7} + 2634 p^{2} T^{8} + 674 p^{3} T^{9} + 45 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 9 T - 673 T^{3} - 3231 T^{4} + 10728 T^{5} + 218523 T^{6} + 10728 p T^{7} - 3231 p^{2} T^{8} - 673 p^{3} T^{9} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 3 T + 123 T^{2} - 279 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 3 T + 6 T^{2} + 549 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 15 T + 120 T^{2} + 621 T^{3} - 2325 T^{4} - 83550 T^{5} - 795413 T^{6} - 83550 p T^{7} - 2325 p^{2} T^{8} + 621 p^{3} T^{9} + 120 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 15 T + 3 T^{2} + 76 T^{3} + 10197 T^{4} - 27621 T^{5} - 489378 T^{6} - 27621 p T^{7} + 10197 p^{2} T^{8} + 76 p^{3} T^{9} + 3 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 9 T + 54 T^{2} + 997 T^{3} + 2313 T^{4} - 2340 T^{5} + 331203 T^{6} - 2340 p T^{7} + 2313 p^{2} T^{8} + 997 p^{3} T^{9} + 54 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 12 T - 78 T^{2} + 522 T^{3} + 14406 T^{4} - 35850 T^{5} - 971597 T^{6} - 35850 p T^{7} + 14406 p^{2} T^{8} + 522 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 30 T + 498 T^{2} - 5187 T^{3} + 498 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 21 T + 375 T^{2} + 3589 T^{3} + 375 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 24 T + 402 T^{2} - 4095 T^{3} + 402 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 9 T - 201 T^{2} + 664 T^{3} + 40017 T^{4} - 71007 T^{5} - 3746098 T^{6} - 71007 p T^{7} + 40017 p^{2} T^{8} + 664 p^{3} T^{9} - 201 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 3 T - 156 T^{2} - 555 T^{3} + 9645 T^{4} + 18822 T^{5} - 624535 T^{6} + 18822 p T^{7} + 9645 p^{2} T^{8} - 555 p^{3} T^{9} - 156 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.48887959514958448919820687881, −6.25542739924107470334920201438, −6.15389398000662034241706129998, −6.03775211735957988400157913614, −5.84045611722474095731263250603, −5.47706293428045336316904736056, −5.45010986472568899679325687920, −5.34986595818804876524319072705, −5.23478329020457030749539254421, −5.21204444987842277841942389622, −4.77985039221304692796907112016, −4.55931138472123728398412753754, −4.43076219582706695934767763619, −3.95057417499698790264056746726, −3.55022963627501722880138021158, −3.48515001727778888681026929082, −2.98094077719765026456036721239, −2.57386335007314017258916863618, −2.36773519230931644309721154281, −2.21902404294144929230327331910, −2.16695683600031131614716884452, −1.89871556772895814121356463839, −1.83054203516174676458594638970, −1.51736990681486145729814613611, −0.61015986673635251661838547009,
0.61015986673635251661838547009, 1.51736990681486145729814613611, 1.83054203516174676458594638970, 1.89871556772895814121356463839, 2.16695683600031131614716884452, 2.21902404294144929230327331910, 2.36773519230931644309721154281, 2.57386335007314017258916863618, 2.98094077719765026456036721239, 3.48515001727778888681026929082, 3.55022963627501722880138021158, 3.95057417499698790264056746726, 4.43076219582706695934767763619, 4.55931138472123728398412753754, 4.77985039221304692796907112016, 5.21204444987842277841942389622, 5.23478329020457030749539254421, 5.34986595818804876524319072705, 5.45010986472568899679325687920, 5.47706293428045336316904736056, 5.84045611722474095731263250603, 6.03775211735957988400157913614, 6.15389398000662034241706129998, 6.25542739924107470334920201438, 6.48887959514958448919820687881
Plot not available for L-functions of degree greater than 10.