| L(s) = 1 | + 16·8-s − 42·11-s − 46·27-s + 138·41-s − 42·43-s − 492·59-s + 64·64-s − 186·67-s − 672·88-s + 438·89-s + 282·97-s + 951·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·8-s − 3.81·11-s − 1.70·27-s + 3.36·41-s − 0.976·43-s − 8.33·59-s + 64-s − 2.77·67-s − 7.63·88-s + 4.92·89-s + 2.90·97-s + 7.85·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06239813207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06239813207\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 3 | \( 1 + 46 T^{3} + 1387 T^{6} + 46 p^{6} T^{9} + p^{12} T^{12} \) |
| good | 5 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 7 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{3}( 1 - 2338 T^{3} + 3694683 T^{6} - 2338 p^{6} T^{9} + p^{12} T^{12} ) \) |
| 13 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 17 | \( ( 1 - 1726 T^{3} - 21158493 T^{6} - 1726 p^{6} T^{9} + p^{12} T^{12} )^{2} \) |
| 19 | \( ( 1 - 2482 T^{3} - 40885557 T^{6} - 2482 p^{6} T^{9} + p^{12} T^{12} )^{2} \) |
| 23 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 29 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 31 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 - p T + p^{2} T^{2} )^{6}( 1 + p T + p^{2} T^{2} )^{6} \) |
| 41 | \( ( 1 - 46 T + 435 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{3}( 1 + 134642 T^{3} + 13378363923 T^{6} + 134642 p^{6} T^{9} + p^{12} T^{12} ) \) |
| 43 | \( ( 1 + 14 T - 1653 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{3}( 1 - 74914 T^{3} - 709255653 T^{6} - 74914 p^{6} T^{9} + p^{12} T^{12} ) \) |
| 47 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 53 | \( ( 1 - p T )^{12}( 1 + p T )^{12} \) |
| 59 | \( ( 1 + 82 T + p^{2} T^{2} )^{6}( 1 + 304958 T^{3} + 50818848123 T^{6} + 304958 p^{6} T^{9} + p^{12} T^{12} ) \) |
| 61 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 67 | \( ( 1 + 62 T - 645 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{3}( 1 - 596626 T^{3} + 265504201707 T^{6} - 596626 p^{6} T^{9} + p^{12} T^{12} ) \) |
| 71 | \( ( 1 - p T + p^{2} T^{2} )^{6}( 1 + p T + p^{2} T^{2} )^{6} \) |
| 73 | \( ( 1 - 593134 T^{3} + 200473715667 T^{6} - 593134 p^{6} T^{9} + p^{12} T^{12} )^{2} \) |
| 79 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )^{2}( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \) |
| 83 | \( ( 1 + 678926 T^{3} + 134000140107 T^{6} + 678926 p^{6} T^{9} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 146 T + p^{2} T^{2} )^{6}( 1 + 146 T + 13395 T^{2} + 146 p^{2} T^{3} + p^{4} T^{4} )^{3} \) |
| 97 | \( ( 1 - 94 T - 573 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{3}( 1 + 1822754 T^{3} + 2489460139587 T^{6} + 1822754 p^{6} T^{9} + p^{12} T^{12} ) \) |
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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
−4.16286924812775611090859001328, −3.98692914527065820882820828697, −3.60075751527311133643485923999, −3.50988510102735851226620486639, −3.39799843259334004474728572376, −3.33882921276189601130612263410, −3.30879107304334293412146578409, −3.20140842967109006895557688265, −3.02262167327000147190432551866, −2.95387136201658022603518078677, −2.76664632987421187016064879012, −2.64434344906429234565388515989, −2.57000834742299159479509153500, −2.07393500300647019319111274207, −2.02673570781524946794090693842, −1.99178623738137250140552640044, −1.94405684849745852986105479750, −1.90742848278950685962180271301, −1.71579074800554632054834364879, −1.38325082770226842212750836696, −1.02003777088971115717093615898, −0.962662162274854992753350357774, −0.57652199949209310578461029466, −0.47046564882534532064779100183, −0.02731170011889083482608911145,
0.02731170011889083482608911145, 0.47046564882534532064779100183, 0.57652199949209310578461029466, 0.962662162274854992753350357774, 1.02003777088971115717093615898, 1.38325082770226842212750836696, 1.71579074800554632054834364879, 1.90742848278950685962180271301, 1.94405684849745852986105479750, 1.99178623738137250140552640044, 2.02673570781524946794090693842, 2.07393500300647019319111274207, 2.57000834742299159479509153500, 2.64434344906429234565388515989, 2.76664632987421187016064879012, 2.95387136201658022603518078677, 3.02262167327000147190432551866, 3.20140842967109006895557688265, 3.30879107304334293412146578409, 3.33882921276189601130612263410, 3.39799843259334004474728572376, 3.50988510102735851226620486639, 3.60075751527311133643485923999, 3.98692914527065820882820828697, 4.16286924812775611090859001328
Plot not available for L-functions of degree greater than 10.
