| L(s) = 1 | + 3·2-s + 3·4-s + 5-s + 2·7-s − 2·8-s + 3·10-s − 11-s − 16·13-s + 6·14-s − 9·16-s − 4·17-s − 3·19-s + 3·20-s − 3·22-s − 7·23-s + 9·25-s − 48·26-s + 6·28-s + 10·29-s + 20·31-s − 9·32-s − 12·34-s + 2·35-s + 3·37-s − 9·38-s − 2·40-s + 12·43-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 0.447·5-s + 0.755·7-s − 0.707·8-s + 0.948·10-s − 0.301·11-s − 4.43·13-s + 1.60·14-s − 9/4·16-s − 0.970·17-s − 0.688·19-s + 0.670·20-s − 0.639·22-s − 1.45·23-s + 9/5·25-s − 9.41·26-s + 1.13·28-s + 1.85·29-s + 3.59·31-s − 1.59·32-s − 2.05·34-s + 0.338·35-s + 0.493·37-s − 1.45·38-s − 0.316·40-s + 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.728459511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.728459511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 52 p T^{7} + 23 p^{2} T^{8} + 17 p^{3} T^{9} - 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + T - 26 T^{2} - 23 T^{3} + 37 p T^{4} + 202 T^{5} - 4853 T^{6} + 202 p T^{7} + 37 p^{3} T^{8} - 23 p^{3} T^{9} - 26 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 8 T + 40 T^{2} + 139 T^{3} + 40 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 4 T - 23 T^{2} - 4 p T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 220 p T^{7} + 410 p^{2} T^{8} - 4 p^{4} T^{9} - 23 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 7 T - 32 T^{2} - 83 T^{3} + 2423 T^{4} + 3946 T^{5} - 46865 T^{6} + 3946 p T^{7} + 2423 p^{2} T^{8} - 83 p^{3} T^{9} - 32 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 5 T + 21 T^{2} + 73 T^{3} + 21 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 20 T + 6 p T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 62234 p T^{7} + 10342 p^{2} T^{8} - 1398 p^{3} T^{9} + 6 p^{5} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \) |
| 41 | \( ( 1 + 90 T^{2} + 9 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 6 T + 60 T^{2} - 389 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 60360 p T^{7} + 13635 p^{2} T^{8} - 33 p^{3} T^{9} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 14 T - 20 T^{2} - 154 T^{3} + 11666 T^{4} + 35126 T^{5} - 499301 T^{6} + 35126 p T^{7} + 11666 p^{2} T^{8} - 154 p^{3} T^{9} - 20 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 13214 p T^{7} + 13762 p^{2} T^{8} + 342 p^{3} T^{9} - 114 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 7 T + 15 T^{2} + 599 T^{3} + 15 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 2 T + 186 T^{2} + 185 T^{3} + 186 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 73314 p T^{7} + 16101 p^{2} T^{8} - 1197 p^{3} T^{9} - 144 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 28 T + 503 T^{2} + 5680 T^{3} + 503 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.08944172263743685607848463060, −4.93908990427110358206093554071, −4.83783695518978570808572298696, −4.53648705930574021266879316607, −4.52159226329993061107549219855, −4.50663271225465705025083967029, −4.47957589547385222976034971406, −4.39083704206111320791882918663, −3.82517997703292050499498052373, −3.78754294841120163808914359849, −3.72093124740181347776476123985, −3.38746376391535684542032113688, −3.01526161670410340705017863544, −2.76960429527432855934885450836, −2.76255528833820574952490688770, −2.67209960011429352830630890602, −2.65467428457696803382658650197, −2.32356884060463520394219276037, −2.15375555088186542980703910146, −1.88715868979233250334510859128, −1.84202883338058939313569372162, −1.23765875919223297204159804616, −0.75969429186910760292145744173, −0.70669708902964603690244121896, −0.34951405689446561372440713971,
0.34951405689446561372440713971, 0.70669708902964603690244121896, 0.75969429186910760292145744173, 1.23765875919223297204159804616, 1.84202883338058939313569372162, 1.88715868979233250334510859128, 2.15375555088186542980703910146, 2.32356884060463520394219276037, 2.65467428457696803382658650197, 2.67209960011429352830630890602, 2.76255528833820574952490688770, 2.76960429527432855934885450836, 3.01526161670410340705017863544, 3.38746376391535684542032113688, 3.72093124740181347776476123985, 3.78754294841120163808914359849, 3.82517997703292050499498052373, 4.39083704206111320791882918663, 4.47957589547385222976034971406, 4.50663271225465705025083967029, 4.52159226329993061107549219855, 4.53648705930574021266879316607, 4.83783695518978570808572298696, 4.93908990427110358206093554071, 5.08944172263743685607848463060
Plot not available for L-functions of degree greater than 10.
