| L(s) = 1 | − 12·4-s + 22·7-s + 43·16-s − 244·19-s − 323·25-s − 264·28-s − 242·31-s − 1.01e3·37-s + 74·43-s − 981·49-s + 1.14e3·61-s + 394·64-s + 2.19e3·67-s + 2.17e3·73-s + 2.92e3·76-s + 1.86e3·79-s + 4.37e3·97-s + 3.87e3·100-s + 3.45e3·103-s − 4.85e3·109-s + 946·112-s − 928·121-s + 2.90e3·124-s + 127-s + 131-s − 5.36e3·133-s + 137-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.18·7-s + 0.671·16-s − 2.94·19-s − 2.58·25-s − 1.78·28-s − 1.40·31-s − 4.52·37-s + 0.262·43-s − 2.86·49-s + 2.40·61-s + 0.769·64-s + 4.00·67-s + 3.48·73-s + 4.41·76-s + 2.65·79-s + 4.57·97-s + 3.87·100-s + 3.30·103-s − 4.26·109-s + 0.798·112-s − 0.697·121-s + 2.10·124-s + 0.000698·127-s + 0.000666·131-s − 3.49·133-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3572444935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3572444935\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3 p^{2} T^{2} + 101 T^{4} + 151 p T^{6} + 31 p^{5} T^{8} + 151 p^{7} T^{10} + 101 p^{12} T^{12} + 3 p^{20} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 + 323 T^{2} + 14109 p T^{4} + 2362538 p T^{6} + 1522988542 T^{8} + 2362538 p^{7} T^{10} + 14109 p^{13} T^{12} + 323 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( ( 1 - 11 T + 96 p T^{2} - 425 T^{3} + 185810 T^{4} - 425 p^{3} T^{5} + 96 p^{7} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( 1 + 928 T^{2} + 4646848 T^{4} + 4183083552 T^{6} + 10154734587870 T^{8} + 4183083552 p^{6} T^{10} + 4646848 p^{12} T^{12} + 928 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 + 9879 T^{2} + 66841397 T^{4} + 413456180966 T^{6} + 2043761199640142 T^{8} + 413456180966 p^{6} T^{10} + 66841397 p^{12} T^{12} + 9879 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 + 122 T + 19262 T^{2} + 1598054 T^{3} + 184664834 T^{4} + 1598054 p^{3} T^{5} + 19262 p^{6} T^{6} + 122 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 + 50984 T^{2} + 1184277008 T^{4} + 17460450114648 T^{6} + 214285541615044382 T^{8} + 17460450114648 p^{6} T^{10} + 1184277008 p^{12} T^{12} + 50984 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( 1 + 55747 T^{2} + 2689371553 T^{4} + 97997970647298 T^{6} + 2493016456293686670 T^{8} + 97997970647298 p^{6} T^{10} + 2689371553 p^{12} T^{12} + 55747 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 + 121 T + 64928 T^{2} + 1397453 T^{3} + 1743770334 T^{4} + 1397453 p^{3} T^{5} + 64928 p^{6} T^{6} + 121 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 + 509 T + 232763 T^{2} + 63937384 T^{3} + 17269851438 T^{4} + 63937384 p^{3} T^{5} + 232763 p^{6} T^{6} + 509 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( 1 + 325831 T^{2} + 55427369189 T^{4} + 6215115063879894 T^{6} + \)\(49\!\cdots\!62\)\( T^{8} + 6215115063879894 p^{6} T^{10} + 55427369189 p^{12} T^{12} + 325831 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( ( 1 - 37 T + 171856 T^{2} + 580529 T^{3} + 16008374778 T^{4} + 580529 p^{3} T^{5} + 171856 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 47 | \( 1 + 673128 T^{2} + 205725969680 T^{4} + 38035116822467736 T^{6} + \)\(47\!\cdots\!38\)\( T^{8} + 38035116822467736 p^{6} T^{10} + 205725969680 p^{12} T^{12} + 673128 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 + 457499 T^{2} + 135151726097 T^{4} + 29356704635913714 T^{6} + \)\(48\!\cdots\!10\)\( T^{8} + 29356704635913714 p^{6} T^{10} + 135151726097 p^{12} T^{12} + 457499 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 + 1197976 T^{2} + 658848403772 T^{4} + 224645430532147560 T^{6} + \)\(53\!\cdots\!30\)\( T^{8} + 224645430532147560 p^{6} T^{10} + 658848403772 p^{12} T^{12} + 1197976 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 - 574 T + 837230 T^{2} - 366110348 T^{3} + 277714385799 T^{4} - 366110348 p^{3} T^{5} + 837230 p^{6} T^{6} - 574 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 1099 T + 1300616 T^{2} - 940063721 T^{3} + 604138073322 T^{4} - 940063721 p^{3} T^{5} + 1300616 p^{6} T^{6} - 1099 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( 1 + 1071656 T^{2} + 748386640784 T^{4} + 370617385955897688 T^{6} + \)\(15\!\cdots\!98\)\( T^{8} + 370617385955897688 p^{6} T^{10} + 748386640784 p^{12} T^{12} + 1071656 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( ( 1 - 1088 T + 1503042 T^{2} - 1067919104 T^{3} + 887747129507 T^{4} - 1067919104 p^{3} T^{5} + 1503042 p^{6} T^{6} - 1088 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 931 T + 2149008 T^{2} - 1338612247 T^{3} + 1623052949342 T^{4} - 1338612247 p^{3} T^{5} + 2149008 p^{6} T^{6} - 931 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 + 3704064 T^{2} + 6275226483968 T^{4} + 6440475301192582208 T^{6} + \)\(44\!\cdots\!02\)\( T^{8} + 6440475301192582208 p^{6} T^{10} + 6275226483968 p^{12} T^{12} + 3704064 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( 1 + 2583560 T^{2} + 3976347469404 T^{4} + 4123207445713521592 T^{6} + \)\(33\!\cdots\!14\)\( T^{8} + 4123207445713521592 p^{6} T^{10} + 3976347469404 p^{12} T^{12} + 2583560 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( ( 1 - 2185 T + 3491318 T^{2} - 4399067099 T^{3} + 4998430507914 T^{4} - 4399067099 p^{3} T^{5} + 3491318 p^{6} T^{6} - 2185 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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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
−3.76988871326615197192834020698, −3.44348856338026429720211744639, −3.32300286958783656611670449928, −3.30591493566974590298148176030, −3.30255224608063172003192291600, −3.00730573986991410855552092354, −2.98439727721299118379224686271, −2.53873256443986689819391695788, −2.36837783298885919606416675926, −2.24726540693446169887959482542, −2.21537667302448073333053445472, −2.09718414329780668360003149511, −2.08666718529677681571590539395, −1.81901921709057545725221894041, −1.70279964804338782269623545252, −1.69169069976669717191970153494, −1.50860122370676696934112211768, −1.50333928784680741044706402142, −0.887138585401610032837980828928, −0.70818017169111831598466452481, −0.63596473011742817013494564085, −0.63033876686140353323652452227, −0.58005491462008896707922207395, −0.15865896336942477923792960432, −0.07004121690060017512610542771,
0.07004121690060017512610542771, 0.15865896336942477923792960432, 0.58005491462008896707922207395, 0.63033876686140353323652452227, 0.63596473011742817013494564085, 0.70818017169111831598466452481, 0.887138585401610032837980828928, 1.50333928784680741044706402142, 1.50860122370676696934112211768, 1.69169069976669717191970153494, 1.70279964804338782269623545252, 1.81901921709057545725221894041, 2.08666718529677681571590539395, 2.09718414329780668360003149511, 2.21537667302448073333053445472, 2.24726540693446169887959482542, 2.36837783298885919606416675926, 2.53873256443986689819391695788, 2.98439727721299118379224686271, 3.00730573986991410855552092354, 3.30255224608063172003192291600, 3.30591493566974590298148176030, 3.32300286958783656611670449928, 3.44348856338026429720211744639, 3.76988871326615197192834020698
Plot not available for L-functions of degree greater than 10.
