| L(s) = 1 | + 4·3-s + 3·4-s − 3·9-s + 12·12-s − 42·13-s − 6·16-s + 84·19-s + 21·25-s − 46·27-s − 9·36-s − 42·37-s − 168·39-s − 156·43-s − 24·48-s − 78·49-s − 126·52-s + 336·57-s + 264·61-s + 42·64-s + 24·67-s + 72·73-s + 84·75-s + 252·76-s − 130·81-s + 126·97-s + 63·100-s + 264·103-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 3/4·4-s − 1/3·9-s + 12-s − 3.23·13-s − 3/8·16-s + 4.42·19-s + 0.839·25-s − 1.70·27-s − 1/4·36-s − 1.13·37-s − 4.30·39-s − 3.62·43-s − 1/2·48-s − 1.59·49-s − 2.42·52-s + 5.89·57-s + 4.32·61-s + 0.656·64-s + 0.358·67-s + 0.986·73-s + 1.11·75-s + 3.31·76-s − 1.60·81-s + 1.29·97-s + 0.629·100-s + 2.56·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.951816276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.951816276\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4 T + 19 T^{2} - 14 p T^{3} + 19 p^{2} T^{4} - 4 p^{4} T^{5} + p^{6} T^{6} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3 T^{2} + 15 T^{4} - 105 T^{6} + 15 p^{4} T^{8} - 3 p^{8} T^{10} + p^{12} T^{12} \) |
| 5 | \( 1 - 21 T^{2} + 1662 T^{4} - 23757 T^{6} + 1662 p^{4} T^{8} - 21 p^{8} T^{10} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 39 T^{2} + 190 T^{3} + 39 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 13 | \( ( 1 + 21 T + 576 T^{2} + 7093 T^{3} + 576 p^{2} T^{4} + 21 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 - 825 T^{2} + 214014 T^{4} - 34165533 T^{6} + 214014 p^{4} T^{8} - 825 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 - 42 T + 1551 T^{2} - 31782 T^{3} + 1551 p^{2} T^{4} - 42 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 2274 T^{2} + 2391951 T^{4} - 1551286560 T^{6} + 2391951 p^{4} T^{8} - 2274 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 1377 T^{2} + 1433106 T^{4} - 1058291885 T^{6} + 1433106 p^{4} T^{8} - 1377 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( ( 1 + 1533 T^{2} - 18334 T^{3} + 1533 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 + 21 T + 3444 T^{2} + 59613 T^{3} + 3444 p^{2} T^{4} + 21 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( 1 - 1881 T^{2} + 5508078 T^{4} - 5453415261 T^{6} + 5508078 p^{4} T^{8} - 1881 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( ( 1 + 78 T + 6423 T^{2} + 288264 T^{3} + 6423 p^{2} T^{4} + 78 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 12174 T^{2} + 63924771 T^{4} - 3939349200 p T^{6} + 63924771 p^{4} T^{8} - 12174 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 3141 T^{2} + 5524830 T^{4} - 9281065133 T^{6} + 5524830 p^{4} T^{8} - 3141 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 - 12654 T^{2} + 79354335 T^{4} - 327947974164 T^{6} + 79354335 p^{4} T^{8} - 12654 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 - 132 T + 16035 T^{2} - 1036920 T^{3} + 16035 p^{2} T^{4} - 132 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( ( 1 - 12 T + 4461 T^{2} + 152074 T^{3} + 4461 p^{2} T^{4} - 12 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 27378 T^{2} + 324883791 T^{4} - 2140235421068 T^{6} + 324883791 p^{4} T^{8} - 27378 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 36 T + 5499 T^{2} - 232868 T^{3} + 5499 p^{2} T^{4} - 36 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( ( 1 + 9093 T^{2} + 356894 T^{3} + 9093 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 83 | \( 1 - 19734 T^{2} + 259216791 T^{4} - 2071634939336 T^{6} + 259216791 p^{4} T^{8} - 19734 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( 1 - 43425 T^{2} + 816079518 T^{4} - 8471737218053 T^{6} + 816079518 p^{4} T^{8} - 43425 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 - 63 T + 7038 T^{2} - 1749359 T^{3} + 7038 p^{2} T^{4} - 63 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−6.07995012249355514675962617752, −5.61355447179225136303298764460, −5.41304635506195144533980762472, −5.32015153243074429639859564518, −5.20188493347255146012569149606, −5.03403383935861415304175324668, −4.96981394118316741435649876338, −4.87461993275927176962925350269, −4.63181018776613383144140167903, −4.16073787700388900582374790560, −3.94523857036193307043372926078, −3.54754392729198218903412400991, −3.52394252062780173723223396877, −3.29339328798050303893008089535, −3.21364218912886372569226107550, −3.06036989820736023005316718634, −2.55967602493741996500640447983, −2.54984186783009059999091093090, −2.27530509611462394229600931876, −2.22496887355170563714154560775, −1.83725739998690612609422489708, −1.32073227337309265802715405113, −1.24666279018163122996734315004, −0.60991896029157343789731929145, −0.26579539377842926067463663120,
0.26579539377842926067463663120, 0.60991896029157343789731929145, 1.24666279018163122996734315004, 1.32073227337309265802715405113, 1.83725739998690612609422489708, 2.22496887355170563714154560775, 2.27530509611462394229600931876, 2.54984186783009059999091093090, 2.55967602493741996500640447983, 3.06036989820736023005316718634, 3.21364218912886372569226107550, 3.29339328798050303893008089535, 3.52394252062780173723223396877, 3.54754392729198218903412400991, 3.94523857036193307043372926078, 4.16073787700388900582374790560, 4.63181018776613383144140167903, 4.87461993275927176962925350269, 4.96981394118316741435649876338, 5.03403383935861415304175324668, 5.20188493347255146012569149606, 5.32015153243074429639859564518, 5.41304635506195144533980762472, 5.61355447179225136303298764460, 6.07995012249355514675962617752
Plot not available for L-functions of degree greater than 10.
