| L(s) = 1 | + 6·2-s − 12·3-s + 12·4-s − 9·5-s − 72·6-s − 21·7-s − 16·8-s + 54·9-s − 54·10-s − 48·11-s − 144·12-s + 57·13-s − 126·14-s + 108·15-s − 144·16-s + 48·17-s + 324·18-s − 282·19-s − 108·20-s + 252·21-s − 288·22-s + 30·23-s + 192·24-s + 333·25-s + 342·26-s − 135·27-s − 252·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 2.30·3-s + 3/2·4-s − 0.804·5-s − 4.89·6-s − 1.13·7-s − 0.707·8-s + 2·9-s − 1.70·10-s − 1.31·11-s − 3.46·12-s + 1.21·13-s − 2.40·14-s + 1.85·15-s − 9/4·16-s + 0.684·17-s + 4.24·18-s − 3.40·19-s − 1.20·20-s + 2.61·21-s − 2.79·22-s + 0.271·23-s + 1.63·24-s + 2.66·25-s + 2.57·26-s − 0.962·27-s − 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04990580188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04990580188\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| 3 | \( 1 + 4 p T + 10 p^{2} T^{2} + 7 p^{4} T^{3} + 10 p^{5} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| good | 5 | \( 1 + 9 T - 252 T^{2} - 153 p T^{3} + 10467 p T^{4} + 17748 T^{5} - 7682339 T^{6} + 17748 p^{3} T^{7} + 10467 p^{7} T^{8} - 153 p^{10} T^{9} - 252 p^{12} T^{10} + 9 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 48 T - 1548 T^{2} - 52818 T^{3} + 4005756 T^{4} + 50678940 T^{5} - 4988044937 T^{6} + 50678940 p^{3} T^{7} + 4005756 p^{6} T^{8} - 52818 p^{9} T^{9} - 1548 p^{12} T^{10} + 48 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 - 57 T - 1509 T^{2} + 144284 T^{3} + 343773 T^{4} - 5530743 p T^{5} - 1052243778 T^{6} - 5530743 p^{4} T^{7} + 343773 p^{6} T^{8} + 144284 p^{9} T^{9} - 1509 p^{12} T^{10} - 57 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 - 24 T + 5220 T^{2} - 78585 T^{3} + 5220 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 141 T + 1029 p T^{2} + 1828069 T^{3} + 1029 p^{4} T^{4} + 141 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 30 T - 27864 T^{2} - 50214 T^{3} + 465205518 T^{4} + 4868880918 T^{5} - 6432386973797 T^{6} + 4868880918 p^{3} T^{7} + 465205518 p^{6} T^{8} - 50214 p^{9} T^{9} - 27864 p^{12} T^{10} - 30 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 + 117 T - 9810 T^{2} - 8104923 T^{3} - 655891011 T^{4} + 63017592930 T^{5} + 36858066204589 T^{6} + 63017592930 p^{3} T^{7} - 655891011 p^{6} T^{8} - 8104923 p^{9} T^{9} - 9810 p^{12} T^{10} + 117 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 - 165 T - 58344 T^{2} + 5495873 T^{3} + 3154650147 T^{4} - 144414314946 T^{5} - 93229664138193 T^{6} - 144414314946 p^{3} T^{7} + 3154650147 p^{6} T^{8} + 5495873 p^{9} T^{9} - 58344 p^{12} T^{10} - 165 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 399 T + 119733 T^{2} + 21982651 T^{3} + 119733 p^{3} T^{4} + 399 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 - 384 T - 98100 T^{2} + 12598554 T^{3} + 22850508924 T^{4} - 1950427947300 T^{5} - 1344970654506389 T^{6} - 1950427947300 p^{3} T^{7} + 22850508924 p^{6} T^{8} + 12598554 p^{9} T^{9} - 98100 p^{12} T^{10} - 384 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 - 771 T + 193974 T^{2} - 50785663 T^{3} + 29084759433 T^{4} - 7431715027980 T^{5} + 1134363686777763 T^{6} - 7431715027980 p^{3} T^{7} + 29084759433 p^{6} T^{8} - 50785663 p^{9} T^{9} + 193974 p^{12} T^{10} - 771 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 + 81 T + 5646 T^{2} + 117759825 T^{3} + 99406605 p T^{4} + 803885546034 T^{5} + 5629933175771815 T^{6} + 803885546034 p^{3} T^{7} + 99406605 p^{7} T^{8} + 117759825 p^{9} T^{9} + 5646 p^{12} T^{10} + 81 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 - 522 T + 190134 T^{2} - 48782187 T^{3} + 190134 p^{3} T^{4} - 522 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 + 21 T - 237354 T^{2} - 181685679 T^{3} + 5748229293 T^{4} + 21030301905816 T^{5} + 14406231366999691 T^{6} + 21030301905816 p^{3} T^{7} + 5748229293 p^{6} T^{8} - 181685679 p^{9} T^{9} - 237354 p^{12} T^{10} + 21 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 780 T - 154002 T^{2} + 55626026 T^{3} + 154589202702 T^{4} - 18121545839646 T^{5} - 34168567588346793 T^{6} - 18121545839646 p^{3} T^{7} + 154589202702 p^{6} T^{8} + 55626026 p^{9} T^{9} - 154002 p^{12} T^{10} - 780 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 384 T - 173424 T^{2} + 85131122 T^{3} - 33547873104 T^{4} + 5955243494544 T^{5} + 20959014584590071 T^{6} + 5955243494544 p^{3} T^{7} - 33547873104 p^{6} T^{8} + 85131122 p^{9} T^{9} - 173424 p^{12} T^{10} - 384 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 57 T + 367377 T^{2} - 182634123 T^{3} + 367377 p^{3} T^{4} - 57 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 117 T + 506133 T^{2} - 134410025 T^{3} + 506133 p^{3} T^{4} + 117 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 - 1383 T - 79770 T^{2} + 85607285 T^{3} + 1022858230077 T^{4} - 5689210514724 p T^{5} - 164486295934003209 T^{6} - 5689210514724 p^{4} T^{7} + 1022858230077 p^{6} T^{8} + 85607285 p^{9} T^{9} - 79770 p^{12} T^{10} - 1383 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 + 12 T - 1053864 T^{2} + 404096754 T^{3} + 510613483716 T^{4} - 216835757815308 T^{5} - 223075525460746457 T^{6} - 216835757815308 p^{3} T^{7} + 510613483716 p^{6} T^{8} + 404096754 p^{9} T^{9} - 1053864 p^{12} T^{10} + 12 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 648 T + 2158218 T^{2} + 892587411 T^{3} + 2158218 p^{3} T^{4} + 648 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 741 T - 1142418 T^{2} + 174586793 T^{3} + 917682825771 T^{4} - 758251433847708 T^{5} - 1257258664172531199 T^{6} - 758251433847708 p^{3} T^{7} + 917682825771 p^{6} T^{8} + 174586793 p^{9} T^{9} - 1142418 p^{12} T^{10} + 741 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−6.77110997061580384931051466387, −6.76032016397542645040898959654, −6.43981894829353877041125884287, −6.09591696342243815366172600726, −6.05339636889922351867868602761, −5.81713895564194447575145964564, −5.53836094782122675244482047884, −5.48717786846838237836845194360, −5.28257771792136302903215146285, −5.03562706260850886473159013910, −4.98678163794366311090939269418, −4.44011402476424208581379670108, −4.30086633793695578849456478803, −4.13817541705663825363878371778, −3.94569915244331703830050933520, −3.63337043255482482911402080751, −3.51494907197545493260224852842, −3.11617224307489224852495705374, −2.60868391804850242218420736133, −2.55959117324735168881248048377, −2.31963662122033386836143188091, −1.55327186300588675897992914475, −0.78723012149814804090427219428, −0.65949936426957856791109234126, −0.05143559332131206523643045035,
0.05143559332131206523643045035, 0.65949936426957856791109234126, 0.78723012149814804090427219428, 1.55327186300588675897992914475, 2.31963662122033386836143188091, 2.55959117324735168881248048377, 2.60868391804850242218420736133, 3.11617224307489224852495705374, 3.51494907197545493260224852842, 3.63337043255482482911402080751, 3.94569915244331703830050933520, 4.13817541705663825363878371778, 4.30086633793695578849456478803, 4.44011402476424208581379670108, 4.98678163794366311090939269418, 5.03562706260850886473159013910, 5.28257771792136302903215146285, 5.48717786846838237836845194360, 5.53836094782122675244482047884, 5.81713895564194447575145964564, 6.05339636889922351867868602761, 6.09591696342243815366172600726, 6.43981894829353877041125884287, 6.76032016397542645040898959654, 6.77110997061580384931051466387
Plot not available for L-functions of degree greater than 10.
