| L(s) = 1 | + 6·2-s − 4-s + 60·8-s + 630·11-s + 869·16-s + 3.78e3·22-s − 216·23-s + 315·25-s + 3.03e3·29-s + 906·32-s − 368·37-s − 4.54e3·43-s − 630·44-s − 1.29e3·46-s + 1.89e3·50-s + 8.29e3·53-s + 1.81e4·58-s − 3.24e3·64-s + 1.88e3·67-s − 1.19e4·71-s − 2.20e3·74-s − 2.02e4·79-s − 2.72e4·86-s + 3.78e4·88-s + 216·92-s − 315·100-s + 4.97e4·106-s + ⋯ |
| L(s) = 1 | + 3/2·2-s − 0.0625·4-s + 0.937·8-s + 5.20·11-s + 3.39·16-s + 7.80·22-s − 0.408·23-s + 0.503·25-s + 3.60·29-s + 0.884·32-s − 0.268·37-s − 2.45·43-s − 0.325·44-s − 0.612·46-s + 0.755·50-s + 2.95·53-s + 5.40·58-s − 0.791·64-s + 0.419·67-s − 2.37·71-s − 0.403·74-s − 3.23·79-s − 3.68·86-s + 4.88·88-s + 0.0255·92-s − 0.0314·100-s + 4.43·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(87.69760756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(87.69760756\) |
| \(L(3)\) |
|
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 | \( ( 1 - 3 T + 7 p T^{2} - 51 p T^{3} + 7 p^{5} T^{4} - 3 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 5 | \( 1 - 63 p T^{2} + 816363 T^{4} - 294525362 T^{6} + 816363 p^{8} T^{8} - 63 p^{17} T^{10} + p^{24} T^{12} \) |
| 11 | \( ( 1 - 315 T + 70931 T^{2} - 9916338 T^{3} + 70931 p^{4} T^{4} - 315 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 13 | \( 1 - 113040 T^{2} + 5984237088 T^{4} - 203996110238930 T^{6} + 5984237088 p^{8} T^{8} - 113040 p^{16} T^{10} + p^{24} T^{12} \) |
| 17 | \( 1 - 87786 T^{2} + 8933211423 T^{4} - 1220139816974444 T^{6} + 8933211423 p^{8} T^{8} - 87786 p^{16} T^{10} + p^{24} T^{12} \) |
| 19 | \( 1 - 641412 T^{2} + 187150089384 T^{4} - 31350646671221918 T^{6} + 187150089384 p^{8} T^{8} - 641412 p^{16} T^{10} + p^{24} T^{12} \) |
| 23 | \( ( 1 + 108 T + 830843 T^{2} + 59556600 T^{3} + 830843 p^{4} T^{4} + 108 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 29 | \( ( 1 - 1515 T + 2099411 T^{2} - 1614430518 T^{3} + 2099411 p^{4} T^{4} - 1515 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 31 | \( 1 - 3519189 T^{2} + 196753558986 p T^{4} - 6770935645876118369 T^{6} + 196753558986 p^{9} T^{8} - 3519189 p^{16} T^{10} + p^{24} T^{12} \) |
| 37 | \( ( 1 + 184 T + 1583192 T^{2} + 2329114882 T^{3} + 1583192 p^{4} T^{4} + 184 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 41 | \( 1 - 14525922 T^{2} + 94031971558383 T^{4} - \)\(34\!\cdots\!92\)\( T^{6} + 94031971558383 p^{8} T^{8} - 14525922 p^{16} T^{10} + p^{24} T^{12} \) |
| 43 | \( ( 1 + 2272 T + 9846248 T^{2} + 13985819182 T^{3} + 9846248 p^{4} T^{4} + 2272 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 47 | \( 1 - 23523150 T^{2} + 245312540183487 T^{4} - \)\(15\!\cdots\!88\)\( T^{6} + 245312540183487 p^{8} T^{8} - 23523150 p^{16} T^{10} + p^{24} T^{12} \) |
| 53 | \( ( 1 - 4149 T + 26233451 T^{2} - 65305136514 T^{3} + 26233451 p^{4} T^{4} - 4149 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 59 | \( 1 - 31400067 T^{2} + 747490121575923 T^{4} - \)\(10\!\cdots\!22\)\( T^{6} + 747490121575923 p^{8} T^{8} - 31400067 p^{16} T^{10} + p^{24} T^{12} \) |
| 61 | \( 1 - 27618246 T^{2} + 165108091404015 T^{4} + \)\(12\!\cdots\!80\)\( T^{6} + 165108091404015 p^{8} T^{8} - 27618246 p^{16} T^{10} + p^{24} T^{12} \) |
| 67 | \( ( 1 - 942 T + 35436948 T^{2} - 45396520472 T^{3} + 35436948 p^{4} T^{4} - 942 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 71 | \( ( 1 + 5988 T + 69445535 T^{2} + 278606834568 T^{3} + 69445535 p^{4} T^{4} + 5988 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 73 | \( 1 - 77384448 T^{2} + 4195503859535568 T^{4} - \)\(13\!\cdots\!30\)\( T^{6} + 4195503859535568 p^{8} T^{8} - 77384448 p^{16} T^{10} + p^{24} T^{12} \) |
| 79 | \( ( 1 + 10101 T + 136782078 T^{2} + 757307270329 T^{3} + 136782078 p^{4} T^{4} + 10101 p^{8} T^{5} + p^{12} T^{6} )^{2} \) |
| 83 | \( 1 - 136283907 T^{2} + 8264939292999435 T^{4} - \)\(38\!\cdots\!26\)\( T^{6} + 8264939292999435 p^{8} T^{8} - 136283907 p^{16} T^{10} + p^{24} T^{12} \) |
| 89 | \( 1 - 299139354 T^{2} + 40876281353101599 T^{4} - \)\(32\!\cdots\!20\)\( T^{6} + 40876281353101599 p^{8} T^{8} - 299139354 p^{16} T^{10} + p^{24} T^{12} \) |
| 97 | \( 1 - 56759355 T^{2} - 577710713629629 T^{4} + \)\(64\!\cdots\!58\)\( T^{6} - 577710713629629 p^{8} T^{8} - 56759355 p^{16} T^{10} + p^{24} T^{12} \) |
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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
−5.29050172107729358330162463120, −4.83073528031723587824786219294, −4.69871146648472440969028680624, −4.69832302202396099630261728511, −4.67683050024681215945449411339, −4.47896677820997572803924783895, −4.38860321025942423103212468096, −3.94734528619354386999647777365, −3.91530446667762353202296032074, −3.74224921424349751939342979095, −3.68102861329786614522059163084, −3.49011550908046115704542046655, −3.23584051520412907880097074664, −2.89485192048078442343042593441, −2.75181628412003468266991495172, −2.55203268316359781349706286176, −1.89062687239752532091584020264, −1.72793702004739845636951514962, −1.69555446070318828082559856475, −1.62217144521265893568088887498, −1.16957607123401211093732731683, −1.05612846124896412825925762864, −0.74598429940511229108929613989, −0.68438539719341961350711558836, −0.41941332189041778810474349360,
0.41941332189041778810474349360, 0.68438539719341961350711558836, 0.74598429940511229108929613989, 1.05612846124896412825925762864, 1.16957607123401211093732731683, 1.62217144521265893568088887498, 1.69555446070318828082559856475, 1.72793702004739845636951514962, 1.89062687239752532091584020264, 2.55203268316359781349706286176, 2.75181628412003468266991495172, 2.89485192048078442343042593441, 3.23584051520412907880097074664, 3.49011550908046115704542046655, 3.68102861329786614522059163084, 3.74224921424349751939342979095, 3.91530446667762353202296032074, 3.94734528619354386999647777365, 4.38860321025942423103212468096, 4.47896677820997572803924783895, 4.67683050024681215945449411339, 4.69832302202396099630261728511, 4.69871146648472440969028680624, 4.83073528031723587824786219294, 5.29050172107729358330162463120
Plot not available for L-functions of degree greater than 10.
