| L(s) = 1 | − 2-s + 7·4-s + 15·5-s + 43·7-s + 6·8-s − 15·10-s + 14·11-s − 40·13-s − 43·14-s + 37·16-s + 332·17-s − 328·19-s + 105·20-s − 14·22-s + 171·23-s + 75·25-s + 40·26-s + 301·28-s − 335·29-s + 352·31-s + 69·32-s − 332·34-s + 645·35-s + 804·37-s + 328·38-s + 90·40-s + 187·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 7/8·4-s + 1.34·5-s + 2.32·7-s + 0.265·8-s − 0.474·10-s + 0.383·11-s − 0.853·13-s − 0.820·14-s + 0.578·16-s + 4.73·17-s − 3.96·19-s + 1.17·20-s − 0.135·22-s + 1.55·23-s + 3/5·25-s + 0.301·26-s + 2.03·28-s − 2.14·29-s + 2.03·31-s + 0.381·32-s − 1.67·34-s + 3.11·35-s + 3.57·37-s + 1.40·38-s + 0.355·40-s + 0.712·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(22.03301793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.03301793\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| good | 2 | \( 1 + T - 3 p T^{2} - 19 T^{3} - 5 p^{2} T^{4} + 43 T^{5} + 737 T^{6} + 43 p^{3} T^{7} - 5 p^{8} T^{8} - 19 p^{9} T^{9} - 3 p^{13} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 43 T + 625 T^{2} - 4310 T^{3} + 39421 T^{4} + 1473581 T^{5} - 66946234 T^{6} + 1473581 p^{3} T^{7} + 39421 p^{6} T^{8} - 4310 p^{9} T^{9} + 625 p^{12} T^{10} - 43 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 14 T - 981 T^{2} + 145154 T^{3} - 1312682 T^{4} - 6914350 p T^{5} + 9056022383 T^{6} - 6914350 p^{4} T^{7} - 1312682 p^{6} T^{8} + 145154 p^{9} T^{9} - 981 p^{12} T^{10} - 14 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 + 40 T - 2539 T^{2} - 35232 T^{3} + 4022098 T^{4} - 118859236 T^{5} - 14072051075 T^{6} - 118859236 p^{3} T^{7} + 4022098 p^{6} T^{8} - 35232 p^{9} T^{9} - 2539 p^{12} T^{10} + 40 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 - 166 T + 23659 T^{2} - 1787440 T^{3} + 23659 p^{3} T^{4} - 166 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 164 T + 27869 T^{2} + 2307068 T^{3} + 27869 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 171 T - 2427 T^{2} + 2784582 T^{3} - 3631569 p T^{4} - 23375770407 T^{5} + 3880984222294 T^{6} - 23375770407 p^{3} T^{7} - 3631569 p^{7} T^{8} + 2784582 p^{9} T^{9} - 2427 p^{12} T^{10} - 171 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 + 335 T + 11727 T^{2} + 1199704 T^{3} + 2210558677 T^{4} + 188505793169 T^{5} - 20536380542074 T^{6} + 188505793169 p^{3} T^{7} + 2210558677 p^{6} T^{8} + 1199704 p^{9} T^{9} + 11727 p^{12} T^{10} + 335 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 - 352 T + 48463 T^{2} - 4331384 T^{3} - 467515598 T^{4} + 431517878828 T^{5} - 103162000220977 T^{6} + 431517878828 p^{3} T^{7} - 467515598 p^{6} T^{8} - 4331384 p^{9} T^{9} + 48463 p^{12} T^{10} - 352 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 - 402 T + 176667 T^{2} - 37389728 T^{3} + 176667 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 - 187 T - 127197 T^{2} + 6771316 T^{3} + 11652659197 T^{4} + 283910950583 T^{5} - 979779052450642 T^{6} + 283910950583 p^{3} T^{7} + 11652659197 p^{6} T^{8} + 6771316 p^{9} T^{9} - 127197 p^{12} T^{10} - 187 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 - 14 p T + 113927 T^{2} + 10980654 T^{3} - 6344547518 T^{4} + 32148434450 T^{5} + 250654741036903 T^{6} + 32148434450 p^{3} T^{7} - 6344547518 p^{6} T^{8} + 10980654 p^{9} T^{9} + 113927 p^{12} T^{10} - 14 p^{16} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 - 665 T + 35475 T^{2} + 10655396 T^{3} + 29735128705 T^{4} - 8179961904035 T^{5} + 54875519418338 T^{6} - 8179961904035 p^{3} T^{7} + 29735128705 p^{6} T^{8} + 10655396 p^{9} T^{9} + 35475 p^{12} T^{10} - 665 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 - 730 T + 552931 T^{2} - 220610956 T^{3} + 552931 p^{3} T^{4} - 730 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 + 298 T - 172689 T^{2} + 87864938 T^{3} + 16154481778 T^{4} - 20738106147710 T^{5} - 438633693159913 T^{6} - 20738106147710 p^{3} T^{7} + 16154481778 p^{6} T^{8} + 87864938 p^{9} T^{9} - 172689 p^{12} T^{10} + 298 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 1439 T + 800471 T^{2} - 410160120 T^{3} + 305155978465 T^{4} - 144780863269729 T^{5} + 52951494992155726 T^{6} - 144780863269729 p^{3} T^{7} + 305155978465 p^{6} T^{8} - 410160120 p^{9} T^{9} + 800471 p^{12} T^{10} - 1439 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 1849 T + 1422835 T^{2} - 1048545668 T^{3} + 941042742781 T^{4} - 583967663605075 T^{5} + 291985089370751786 T^{6} - 583967663605075 p^{3} T^{7} + 941042742781 p^{6} T^{8} - 1048545668 p^{9} T^{9} + 1422835 p^{12} T^{10} - 1849 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 70 T + 388273 T^{2} - 173667512 T^{3} + 388273 p^{3} T^{4} + 70 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 368 T + 794123 T^{2} + 151388768 T^{3} + 794123 p^{3} T^{4} + 368 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 - 382 T - 945269 T^{2} + 59882698 T^{3} + 561648238738 T^{4} + 58077227393690 T^{5} - 326016591815072533 T^{6} + 58077227393690 p^{3} T^{7} + 561648238738 p^{6} T^{8} + 59882698 p^{9} T^{9} - 945269 p^{12} T^{10} - 382 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 + 831 T + 136173 T^{2} + 471764082 T^{3} - 5047186575 T^{4} - 258220893966909 T^{5} - 27597615887273402 T^{6} - 258220893966909 p^{3} T^{7} - 5047186575 p^{6} T^{8} + 471764082 p^{9} T^{9} + 136173 p^{12} T^{10} + 831 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 1719 T + 2353398 T^{2} + 2298177027 T^{3} + 2353398 p^{3} T^{4} + 1719 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 282 T - 2590827 T^{2} + 242659634 T^{3} + 4553223700122 T^{4} - 189773809219314 T^{5} - 4805990536587810771 T^{6} - 189773809219314 p^{3} T^{7} + 4553223700122 p^{6} T^{8} + 242659634 p^{9} T^{9} - 2590827 p^{12} T^{10} - 282 p^{15} T^{11} + p^{18} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.79322936238498789657160186620, −6.74140541695804851450080587645, −6.57032579213853760814771673729, −6.15916798433336867799682743872, −5.85390930300282892862629358172, −5.75205965696564852229317901696, −5.63782406773366193130652842118, −5.45216812268642077214889883797, −5.30936765784730685166411725727, −5.20819166132410274418430423085, −4.53586339337128388306681045214, −4.40490300439431095192658962542, −4.17291353941390207178393797955, −4.16676995840578953738506014239, −3.82378101477266373673975012897, −3.51366797203490508322428768076, −2.84651664562852260055261528008, −2.54482508710635078438093378137, −2.46606726067984353045417699574, −2.27852418771131265195829680952, −2.09778957645580124203290300113, −1.28117760485720698467757558440, −1.23716402281219157545915361423, −0.982715769674182569809337261981, −0.77680866648430170685307633238,
0.77680866648430170685307633238, 0.982715769674182569809337261981, 1.23716402281219157545915361423, 1.28117760485720698467757558440, 2.09778957645580124203290300113, 2.27852418771131265195829680952, 2.46606726067984353045417699574, 2.54482508710635078438093378137, 2.84651664562852260055261528008, 3.51366797203490508322428768076, 3.82378101477266373673975012897, 4.16676995840578953738506014239, 4.17291353941390207178393797955, 4.40490300439431095192658962542, 4.53586339337128388306681045214, 5.20819166132410274418430423085, 5.30936765784730685166411725727, 5.45216812268642077214889883797, 5.63782406773366193130652842118, 5.75205965696564852229317901696, 5.85390930300282892862629358172, 6.15916798433336867799682743872, 6.57032579213853760814771673729, 6.74140541695804851450080587645, 6.79322936238498789657160186620
Plot not available for L-functions of degree greater than 10.
