| L(s) = 1 | − 3·5-s + 6·7-s − 3·9-s + 3·11-s − 9·17-s + 3·19-s + 15·23-s + 3·25-s − 3·27-s + 6·29-s + 3·31-s − 18·35-s − 6·37-s − 9·41-s + 21·43-s + 9·45-s + 12·47-s + 21·49-s − 9·53-s − 9·55-s + 3·59-s + 24·61-s − 18·63-s − 21·71-s − 3·73-s + 18·77-s + 6·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 2.26·7-s − 9-s + 0.904·11-s − 2.18·17-s + 0.688·19-s + 3.12·23-s + 3/5·25-s − 0.577·27-s + 1.11·29-s + 0.538·31-s − 3.04·35-s − 0.986·37-s − 1.40·41-s + 3.20·43-s + 1.34·45-s + 1.75·47-s + 3·49-s − 1.23·53-s − 1.21·55-s + 0.390·59-s + 3.07·61-s − 2.26·63-s − 2.49·71-s − 0.351·73-s + 2.05·77-s + 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.378224365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.378224365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 12 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 6 T + 15 T^{2} - 27 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 302 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 15 T + 135 T^{2} - 766 T^{3} + 135 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 63 T^{2} - 201 T^{3} + 63 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 52 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 348 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 78 T^{2} + 616 T^{3} + 78 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 21 T + 222 T^{2} - 1622 T^{3} + 222 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 81 T^{2} - 456 T^{3} + 81 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 502 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 153 T^{2} - 290 T^{3} + 153 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 24 T + 339 T^{2} - 3120 T^{3} + 339 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 105 T^{2} - 123 T^{3} + 105 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 21 T + 216 T^{2} + 1676 T^{3} + 216 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 213 T^{2} + 426 T^{3} + 213 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 4 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 33 T + 594 T^{2} - 6582 T^{3} + 594 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 231 T^{2} - 924 T^{3} + 231 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 195 T^{2} - 2072 T^{3} + 195 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72628162876852593641667557946, −7.47371111524425191184621650791, −7.07567106228648410087452310696, −6.98890758483376570007619622358, −6.65875021716116288090863418964, −6.64265791188034580056217829090, −6.10570412018116567052355136848, −5.71190213118998710536517377098, −5.61747085790768619052106141947, −5.09824761944150615794418044588, −4.99110172456217632628585428590, −4.91677917679657060362896309436, −4.41151625604023429027447590391, −4.30402067798938240792304427593, −4.07709488284577133438182326182, −3.75019801656087253981055009982, −3.29221799126032696049378261547, −3.12516300103931459250335527569, −2.65898821504962878556868524179, −2.33974577060600775093024291230, −2.15005705055777018504567760222, −1.64620270307687282364260057995, −1.12388712759297604209923031425, −0.845390758275827836033921617363, −0.52206988421321736225961631490,
0.52206988421321736225961631490, 0.845390758275827836033921617363, 1.12388712759297604209923031425, 1.64620270307687282364260057995, 2.15005705055777018504567760222, 2.33974577060600775093024291230, 2.65898821504962878556868524179, 3.12516300103931459250335527569, 3.29221799126032696049378261547, 3.75019801656087253981055009982, 4.07709488284577133438182326182, 4.30402067798938240792304427593, 4.41151625604023429027447590391, 4.91677917679657060362896309436, 4.99110172456217632628585428590, 5.09824761944150615794418044588, 5.61747085790768619052106141947, 5.71190213118998710536517377098, 6.10570412018116567052355136848, 6.64265791188034580056217829090, 6.65875021716116288090863418964, 6.98890758483376570007619622358, 7.07567106228648410087452310696, 7.47371111524425191184621650791, 7.72628162876852593641667557946
