| L(s) = 1 | − 6·2-s + 13·4-s − 6·8-s − 22·16-s − 30·25-s − 30·29-s + 24·32-s + 2·37-s − 10·43-s + 180·50-s − 12·53-s + 180·58-s + 63·64-s + 12·67-s − 12·74-s − 6·79-s + 60·86-s − 390·100-s + 72·106-s + 90·107-s − 34·109-s + 108·113-s − 390·116-s + 88·121-s + 127-s − 180·128-s + 131-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 13/2·4-s − 2.12·8-s − 5.5·16-s − 6·25-s − 5.57·29-s + 4.24·32-s + 0.328·37-s − 1.52·43-s + 25.4·50-s − 1.64·53-s + 23.6·58-s + 63/8·64-s + 1.46·67-s − 1.39·74-s − 0.675·79-s + 6.46·86-s − 39·100-s + 6.99·106-s + 8.70·107-s − 3.25·109-s + 10.1·113-s − 36.2·116-s + 8·121-s + 0.0887·127-s − 15.9·128-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06246823838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06246823838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + 31 T^{6} + 3 p^{4} T^{7} + 17 p^{2} T^{8} + 3 p^{5} T^{9} + 7 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( ( 1 + 3 p T^{2} + 141 T^{4} + 829 T^{6} + 141 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 4 p T^{2} + 8 p^{2} T^{4} - 13307 T^{6} + 8 p^{4} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( 1 + 33 T^{2} + 27 p T^{4} + 2144 T^{6} + 53457 T^{8} + 906351 T^{10} + 9826878 T^{12} + 906351 p^{2} T^{14} + 53457 p^{4} T^{16} + 2144 p^{6} T^{18} + 27 p^{9} T^{20} + 33 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( 1 - 48 T^{2} + 966 T^{4} - 10534 T^{6} + 115002 T^{8} - 2806146 T^{10} + 59116263 T^{12} - 2806146 p^{2} T^{14} + 115002 p^{4} T^{16} - 10534 p^{6} T^{18} + 966 p^{8} T^{20} - 48 p^{10} T^{22} + p^{12} T^{24} \) |
| 19 | \( 1 + 51 T^{2} + 1002 T^{4} + 10607 T^{6} + 170541 T^{8} + 4486122 T^{10} + 90313521 T^{12} + 4486122 p^{2} T^{14} + 170541 p^{4} T^{16} + 10607 p^{6} T^{18} + 1002 p^{8} T^{20} + 51 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( ( 1 - 116 T^{2} + 6008 T^{4} - 177947 T^{6} + 6008 p^{2} T^{8} - 116 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 15 T + 184 T^{2} + 1635 T^{3} + 13205 T^{4} + 85158 T^{5} + 500899 T^{6} + 85158 p T^{7} + 13205 p^{2} T^{8} + 1635 p^{3} T^{9} + 184 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( 1 + 57 T^{2} + 3216 T^{4} + 97787 T^{6} + 2702697 T^{8} + 47442966 T^{10} + 1072931313 T^{12} + 47442966 p^{2} T^{14} + 2702697 p^{4} T^{16} + 97787 p^{6} T^{18} + 3216 p^{8} T^{20} + 57 p^{10} T^{22} + p^{12} T^{24} \) |
| 37 | \( ( 1 - T - 106 T^{2} + 39 T^{3} + 7417 T^{4} - 1262 T^{5} - 318347 T^{6} - 1262 p T^{7} + 7417 p^{2} T^{8} + 39 p^{3} T^{9} - 106 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 41 | \( 1 - 174 T^{2} + 15930 T^{4} - 1039210 T^{6} + 54933612 T^{8} - 2550462528 T^{10} + 108692576859 T^{12} - 2550462528 p^{2} T^{14} + 54933612 p^{4} T^{16} - 1039210 p^{6} T^{18} + 15930 p^{8} T^{20} - 174 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 + 5 T - 58 T^{2} - 51 T^{3} + 2155 T^{4} - 7106 T^{5} - 129149 T^{6} - 7106 p T^{7} + 2155 p^{2} T^{8} - 51 p^{3} T^{9} - 58 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 - 189 T^{2} + 18726 T^{4} - 1271869 T^{6} + 67940493 T^{8} - 3218131254 T^{10} + 150306249081 T^{12} - 3218131254 p^{2} T^{14} + 67940493 p^{4} T^{16} - 1271869 p^{6} T^{18} + 18726 p^{8} T^{20} - 189 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( ( 1 + 6 T + 118 T^{2} + 12 p T^{3} + 6392 T^{4} + 456 p T^{5} + 313969 T^{6} + 456 p^{2} T^{7} + 6392 p^{2} T^{8} + 12 p^{4} T^{9} + 118 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 59 | \( 1 - 135 T^{2} + 4578 T^{4} + 47801 T^{6} + 11150547 T^{8} - 1706849076 T^{10} + 108381578457 T^{12} - 1706849076 p^{2} T^{14} + 11150547 p^{4} T^{16} + 47801 p^{6} T^{18} + 4578 p^{8} T^{20} - 135 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( 1 + 342 T^{2} + 66840 T^{4} + 9095042 T^{6} + 949804386 T^{8} + 78844563978 T^{10} + 5324903072391 T^{12} + 78844563978 p^{2} T^{14} + 949804386 p^{4} T^{16} + 9095042 p^{6} T^{18} + 66840 p^{8} T^{20} + 342 p^{10} T^{22} + p^{12} T^{24} \) |
| 67 | \( ( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 28236 p T^{7} + 17268 p^{2} T^{8} + 506 p^{3} T^{9} - 150 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 263 T^{2} + 31517 T^{4} - 2539307 T^{6} + 31517 p^{2} T^{8} - 263 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( 1 + 159 T^{2} + 14718 T^{4} + 5927 p T^{6} - 43837515 T^{8} - 6799754718 T^{10} - 621204909015 T^{12} - 6799754718 p^{2} T^{14} - 43837515 p^{4} T^{16} + 5927 p^{7} T^{18} + 14718 p^{8} T^{20} + 159 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 + 3 T - 204 T^{2} - 151 T^{3} + 27357 T^{4} + 24 p T^{5} - 31623 p T^{6} + 24 p^{2} T^{7} + 27357 p^{2} T^{8} - 151 p^{3} T^{9} - 204 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - 264 T^{2} + 27306 T^{4} - 2832334 T^{6} + 401548758 T^{8} - 35501027934 T^{10} + 2433955301391 T^{12} - 35501027934 p^{2} T^{14} + 401548758 p^{4} T^{16} - 2832334 p^{6} T^{18} + 27306 p^{8} T^{20} - 264 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( 1 - 210 T^{2} + 12198 T^{4} - 238954 T^{6} + 86692932 T^{8} - 10094552136 T^{10} + 599935426827 T^{12} - 10094552136 p^{2} T^{14} + 86692932 p^{4} T^{16} - 238954 p^{6} T^{18} + 12198 p^{8} T^{20} - 210 p^{10} T^{22} + p^{12} T^{24} \) |
| 97 | \( 1 + 513 T^{2} + 147894 T^{4} + 30072485 T^{6} + 4748892315 T^{8} + 607745530428 T^{10} + 64510991774769 T^{12} + 607745530428 p^{2} T^{14} + 4748892315 p^{4} T^{16} + 30072485 p^{6} T^{18} + 147894 p^{8} T^{20} + 513 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.17097235918170767507157389149, −3.07389547093506200895876677309, −2.64153832684846225547251553185, −2.55241998963393461414333501986, −2.48607716739405860820816413414, −2.40717208591725041042839976977, −2.28306764018306707317121606651, −2.14787020452562483438412324776, −2.13352302990975495112984355505, −2.04920487706899171874631003106, −1.95073447655321121322462559025, −1.86858803126071739280048903086, −1.68838736129483826167143237013, −1.65194108415272997597957898185, −1.60262062401629105918000610695, −1.52439397906601071258412321713, −1.49227346147455446249620186889, −0.981933301990002622881218842672, −0.937101782633242955446360987957, −0.901715302209949352100585708674, −0.54143477192856033064330157721, −0.51215733528332336437506961011, −0.32127010944374408266134964751, −0.26684101023497472966883172841, −0.21872607722923654901599293966,
0.21872607722923654901599293966, 0.26684101023497472966883172841, 0.32127010944374408266134964751, 0.51215733528332336437506961011, 0.54143477192856033064330157721, 0.901715302209949352100585708674, 0.937101782633242955446360987957, 0.981933301990002622881218842672, 1.49227346147455446249620186889, 1.52439397906601071258412321713, 1.60262062401629105918000610695, 1.65194108415272997597957898185, 1.68838736129483826167143237013, 1.86858803126071739280048903086, 1.95073447655321121322462559025, 2.04920487706899171874631003106, 2.13352302990975495112984355505, 2.14787020452562483438412324776, 2.28306764018306707317121606651, 2.40717208591725041042839976977, 2.48607716739405860820816413414, 2.55241998963393461414333501986, 2.64153832684846225547251553185, 3.07389547093506200895876677309, 3.17097235918170767507157389149
Plot not available for L-functions of degree greater than 10.
