| L(s) = 1 | − 6·2-s + 12·4-s + 9·5-s − 21·7-s + 16·8-s − 54·10-s + 48·11-s + 57·13-s + 126·14-s − 144·16-s − 48·17-s − 282·19-s + 108·20-s − 288·22-s − 30·23-s + 333·25-s − 342·26-s − 252·28-s + 117·29-s + 165·31-s + 288·32-s + 288·34-s − 189·35-s − 798·37-s + 1.69e3·38-s + 144·40-s − 384·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 0.804·5-s − 1.13·7-s + 0.707·8-s − 1.70·10-s + 1.31·11-s + 1.21·13-s + 2.40·14-s − 9/4·16-s − 0.684·17-s − 3.40·19-s + 1.20·20-s − 2.79·22-s − 0.271·23-s + 2.66·25-s − 2.57·26-s − 1.70·28-s + 0.749·29-s + 0.955·31-s + 1.59·32-s + 1.45·34-s − 0.912·35-s − 3.54·37-s + 7.22·38-s + 0.569·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.074830368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074830368\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 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
−5.88030614977216751502045408771, −5.44175509787998942656364000648, −5.44156174308949724819363670400, −5.10455874635437675444708178112, −4.86029635589169056788032899548, −4.74785996558802117288709530860, −4.74502202688261120882019203940, −4.35002722708563737073948210630, −3.94084682723032223785564123231, −3.90808389141716625258768706525, −3.88313329092733409783581318214, −3.76574156844207989967976879858, −3.19186679419462455334543807022, −3.07109258108179013060951715781, −2.88950341552374098191092817665, −2.52193079765936016328513616667, −2.30366209190196720595401369671, −2.04102474837304624602803156967, −1.76533987319459201979132227069, −1.64225872338389418438839086302, −1.25456201988377070687392844048, −1.12770369256774072449370671204, −0.67859998068754003579610327201, −0.36691392597921359581767233521, −0.30765300743528369465650107125,
0.30765300743528369465650107125, 0.36691392597921359581767233521, 0.67859998068754003579610327201, 1.12770369256774072449370671204, 1.25456201988377070687392844048, 1.64225872338389418438839086302, 1.76533987319459201979132227069, 2.04102474837304624602803156967, 2.30366209190196720595401369671, 2.52193079765936016328513616667, 2.88950341552374098191092817665, 3.07109258108179013060951715781, 3.19186679419462455334543807022, 3.76574156844207989967976879858, 3.88313329092733409783581318214, 3.90808389141716625258768706525, 3.94084682723032223785564123231, 4.35002722708563737073948210630, 4.74502202688261120882019203940, 4.74785996558802117288709530860, 4.86029635589169056788032899548, 5.10455874635437675444708178112, 5.44156174308949724819363670400, 5.44175509787998942656364000648, 5.88030614977216751502045408771
Plot not available for L-functions of degree greater than 10.
