| L(s) = 1 | + 80·7-s − 120·13-s − 448·31-s − 600·37-s − 480·43-s + 3.20e3·49-s − 528·61-s − 2.08e3·67-s − 2.60e3·73-s + 882·81-s − 9.60e3·91-s + 120·97-s − 6.80e3·103-s + 488·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.20e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4.31·7-s − 2.56·13-s − 2.59·31-s − 2.66·37-s − 1.70·43-s + 9.32·49-s − 1.10·61-s − 3.79·67-s − 4.16·73-s + 1.20·81-s − 11.0·91-s + 0.125·97-s − 6.50·103-s + 0.366·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.27·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05342261183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05342261183\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 98 p^{2} T^{4} + p^{12} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 40 T + 800 T^{2} - 18600 T^{3} + 417566 T^{4} - 18600 p^{3} T^{5} + 800 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 244 T^{2} - 2681994 T^{4} - 244 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 + 60 T + 1800 T^{2} + 140100 T^{3} + 10885406 T^{4} + 140100 p^{3} T^{5} + 1800 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 17 | \( 1 - 9747844 T^{4} + 261455638761606 T^{8} - 9747844 p^{12} T^{12} + p^{24} T^{16} \) |
| 19 | \( ( 1 - 14228 T^{2} + 102518358 T^{4} - 14228 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 - 180861124 T^{4} + 11935037592636486 T^{8} - 180861124 p^{12} T^{12} + p^{24} T^{16} \) |
| 29 | \( ( 1 + 24076 T^{2} + 111520086 T^{4} + 24076 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 112 T + 318 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 + 300 T + 45000 T^{2} + 10989300 T^{3} + 2487043838 T^{4} + 10989300 p^{3} T^{5} + 45000 p^{6} T^{6} + 300 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 - 75364 T^{2} + 5545926 p^{2} T^{4} - 75364 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 + 240 T + 28800 T^{2} - 4818000 T^{3} - 9110563474 T^{4} - 4818000 p^{3} T^{5} + 28800 p^{6} T^{6} + 240 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( 1 - 26528498564 T^{4} + \)\(40\!\cdots\!06\)\( T^{8} - 26528498564 p^{12} T^{12} + p^{24} T^{16} \) |
| 53 | \( 1 + 4859555036 T^{4} + \)\(92\!\cdots\!06\)\( T^{8} + 4859555036 p^{12} T^{12} + p^{24} T^{16} \) |
| 59 | \( ( 1 + 359798 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 + 132 T + 395918 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 + 1040 T + 540800 T^{2} + 446830800 T^{3} + 352579082126 T^{4} + 446830800 p^{3} T^{5} + 540800 p^{6} T^{6} + 1040 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1354564 T^{2} + 714755475366 T^{4} - 1354564 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 1300 T + 845000 T^{2} + 778724700 T^{3} + 673546684718 T^{4} + 778724700 p^{3} T^{5} + 845000 p^{6} T^{6} + 1300 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 968164 T^{2} + 562756853766 T^{4} - 968164 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 372870495922 T^{4} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 2362396 T^{2} + 2378183937126 T^{4} + 2362396 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 60 T + 1800 T^{2} - 53589300 T^{3} + 1595070755726 T^{4} - 53589300 p^{3} T^{5} + 1800 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.75334376923432323877609152167, −4.73106370572971780359611327702, −4.71753454978360751367925804239, −4.35717310663215347453402288052, −4.29017593224253066326531886712, −3.96963455015863636863921453765, −3.80749292283401733927963654150, −3.71517024879440658076541338014, −3.59928586620295000600533961008, −3.50015811464572358556210632694, −2.97877569563641491384780608568, −2.79002377038801377586391372729, −2.70502939450142479380407496897, −2.66105101402405991501625911839, −2.40219995706526278395066669336, −2.25402333324226769970919401917, −1.71354228562094563885473594823, −1.63398600593029946727143234817, −1.62349181433226999785118407771, −1.60268726245951923274816970463, −1.42866012678474453317581883285, −1.23273834143135468784298771161, −0.65996559706368681621651634324, −0.16863813403455893851247804889, −0.03971544617748549215323007172,
0.03971544617748549215323007172, 0.16863813403455893851247804889, 0.65996559706368681621651634324, 1.23273834143135468784298771161, 1.42866012678474453317581883285, 1.60268726245951923274816970463, 1.62349181433226999785118407771, 1.63398600593029946727143234817, 1.71354228562094563885473594823, 2.25402333324226769970919401917, 2.40219995706526278395066669336, 2.66105101402405991501625911839, 2.70502939450142479380407496897, 2.79002377038801377586391372729, 2.97877569563641491384780608568, 3.50015811464572358556210632694, 3.59928586620295000600533961008, 3.71517024879440658076541338014, 3.80749292283401733927963654150, 3.96963455015863636863921453765, 4.29017593224253066326531886712, 4.35717310663215347453402288052, 4.71753454978360751367925804239, 4.73106370572971780359611327702, 4.75334376923432323877609152167
Plot not available for L-functions of degree greater than 10.
