L(s) = 1 | − 3·2-s + 6·4-s − 6·5-s − 9·8-s + 18·10-s + 12·11-s − 3·13-s + 12·16-s + 6·17-s − 3·19-s − 36·20-s − 36·22-s + 24·23-s − 3·25-s + 9·26-s − 9·27-s − 9·29-s − 3·31-s − 12·32-s − 18·34-s + 3·37-s + 9·38-s + 54·40-s + 3·43-s + 72·44-s − 72·46-s + 3·47-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 3·4-s − 2.68·5-s − 3.18·8-s + 5.69·10-s + 3.61·11-s − 0.832·13-s + 3·16-s + 1.45·17-s − 0.688·19-s − 8.04·20-s − 7.67·22-s + 5.00·23-s − 3/5·25-s + 1.76·26-s − 1.73·27-s − 1.67·29-s − 0.538·31-s − 2.12·32-s − 3.08·34-s + 0.493·37-s + 1.45·38-s + 8.53·40-s + 0.457·43-s + 10.8·44-s − 10.6·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6859937569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6859937569\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + p^{2} T^{3} + p^{3} T^{6} \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 3 T + 3 T^{2} - 3 T^{4} - 3 p T^{5} - 11 T^{6} - 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( ( 1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 - 6 T + 42 T^{2} - 135 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 3 T + 3 T^{2} + 76 T^{3} + 45 T^{4} - 135 T^{5} + 3246 T^{6} - 135 p T^{7} + 45 p^{2} T^{8} + 76 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 6 T - 24 T^{2} + 54 T^{3} + 1338 T^{4} - 1914 T^{5} - 18929 T^{6} - 1914 p T^{7} + 1338 p^{2} T^{8} + 54 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T - 42 T^{2} - 41 T^{3} + 1341 T^{4} + 216 T^{5} - 29541 T^{6} + 216 p T^{7} + 1341 p^{2} T^{8} - 41 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 12 T + 96 T^{2} - 549 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 9414 p T^{7} - 579 p^{2} T^{8} + 81 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 3 T - 6 T^{2} + 319 T^{3} + 171 T^{4} - 1962 T^{5} + 62727 T^{6} - 1962 p T^{7} + 171 p^{2} T^{8} + 319 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 3 T - 78 T^{2} + 405 T^{3} + 2481 T^{4} - 11064 T^{5} - 57089 T^{6} - 11064 p T^{7} + 2481 p^{2} T^{8} + 405 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 6 T - 114 T^{2} - 378 T^{3} + 10716 T^{4} + 17304 T^{5} - 587549 T^{6} + 17304 p T^{7} + 10716 p^{2} T^{8} - 378 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 3 T - 96 T^{2} - 495 T^{3} + 3615 T^{4} + 15798 T^{5} - 107021 T^{6} + 15798 p T^{7} + 3615 p^{2} T^{8} - 495 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T - 132 T^{2} + 418 T^{3} + 13698 T^{4} - 19134 T^{5} - 893289 T^{6} - 19134 p T^{7} + 13698 p^{2} T^{8} + 418 p^{3} T^{9} - 132 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 50094 p T^{7} + 15318 p^{2} T^{8} + 518 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 21 T + 138 T^{2} + 769 T^{3} + 10953 T^{4} + 30402 T^{5} - 450903 T^{6} + 30402 p T^{7} + 10953 p^{2} T^{8} + 769 p^{3} T^{9} + 138 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 195678 p T^{7} + 25767 p^{2} T^{8} - 499 p^{3} T^{9} + 84 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 12 T - 60 T^{2} + 198 T^{3} + 7584 T^{4} + 70800 T^{5} - 1684181 T^{6} + 70800 p T^{7} + 7584 p^{2} T^{8} + 198 p^{3} T^{9} - 60 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 3 T - 114 T^{2} - 149 T^{3} + 2421 T^{4} - 11502 T^{5} + 340233 T^{6} - 11502 p T^{7} + 2421 p^{2} T^{8} - 149 p^{3} T^{9} - 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.19693244313385338104612223364, −5.90343120031207152084655717913, −5.64009487632801313532923384073, −5.47414082987462139635001727830, −5.42782652800769948317404003880, −5.24853283800690914433769717941, −4.97936377199446691309229423952, −4.65415282673417634060203467985, −4.27457840028778284698684265837, −4.14746444960558267365807583154, −4.14332904254177473246108783603, −3.96567197639062972090645501305, −3.77321975279774710630389761234, −3.47539630486383945661475046234, −3.36778734311101839185545425651, −3.30322834398127462867770409850, −2.79722891893787142943577745980, −2.79217243020816888945773277591, −2.16362808756966934270816126251, −1.93089288537058695160671753264, −1.80164826984576420069899669777, −1.33635310078923381695324664184, −1.23454846392654247227327934858, −0.64212283007056521566962026463, −0.49314541248281505733290936560,
0.49314541248281505733290936560, 0.64212283007056521566962026463, 1.23454846392654247227327934858, 1.33635310078923381695324664184, 1.80164826984576420069899669777, 1.93089288537058695160671753264, 2.16362808756966934270816126251, 2.79217243020816888945773277591, 2.79722891893787142943577745980, 3.30322834398127462867770409850, 3.36778734311101839185545425651, 3.47539630486383945661475046234, 3.77321975279774710630389761234, 3.96567197639062972090645501305, 4.14332904254177473246108783603, 4.14746444960558267365807583154, 4.27457840028778284698684265837, 4.65415282673417634060203467985, 4.97936377199446691309229423952, 5.24853283800690914433769717941, 5.42782652800769948317404003880, 5.47414082987462139635001727830, 5.64009487632801313532923384073, 5.90343120031207152084655717913, 6.19693244313385338104612223364
Plot not available for L-functions of degree greater than 10.