| L(s) = 1 | + 6·3-s + 4·7-s + 21·9-s − 11-s + 2·13-s + 6·17-s − 3·19-s + 24·21-s + 23-s + 56·27-s + 10·29-s + 6·31-s − 6·33-s + 14·37-s + 12·39-s + 2·41-s + 9·43-s + 11·47-s − 6·49-s + 36·51-s − 9·53-s − 18·57-s + 6·59-s + 10·61-s + 84·63-s + 13·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 1.51·7-s + 7·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.688·19-s + 5.23·21-s + 0.208·23-s + 10.7·27-s + 1.85·29-s + 1.07·31-s − 1.04·33-s + 2.30·37-s + 1.92·39-s + 0.312·41-s + 1.37·43-s + 1.60·47-s − 6/7·49-s + 5.04·51-s − 1.23·53-s − 2.38·57-s + 0.781·59-s + 1.28·61-s + 10.5·63-s + 1.58·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(262.4852424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(262.4852424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( 1 \) |
| 31 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 4 T + 22 T^{2} - 68 T^{3} + 255 T^{4} - 688 T^{5} + 2052 T^{6} - 688 p T^{7} + 255 p^{2} T^{8} - 68 p^{3} T^{9} + 22 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + T + 19 T^{2} + 64 T^{3} + 361 T^{4} + 7 p^{2} T^{5} + 5106 T^{6} + 7 p^{3} T^{7} + 361 p^{2} T^{8} + 64 p^{3} T^{9} + 19 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 2 T + 32 T^{2} - 36 T^{3} + 415 T^{4} + 350 T^{5} + 4336 T^{6} + 350 p T^{7} + 415 p^{2} T^{8} - 36 p^{3} T^{9} + 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 6 T + 59 T^{2} - 278 T^{3} + 2010 T^{4} - 7458 T^{5} + 39939 T^{6} - 7458 p T^{7} + 2010 p^{2} T^{8} - 278 p^{3} T^{9} + 59 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T + 27 T^{2} + 6 p T^{3} + 801 T^{4} + 2055 T^{5} + 15914 T^{6} + 2055 p T^{7} + 801 p^{2} T^{8} + 6 p^{4} T^{9} + 27 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - T + 55 T^{2} - 52 T^{3} + 1477 T^{4} - 907 T^{5} + 30798 T^{6} - 907 p T^{7} + 1477 p^{2} T^{8} - 52 p^{3} T^{9} + 55 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 10 T + 121 T^{2} - 582 T^{3} + 3918 T^{4} - 8714 T^{5} + 78225 T^{6} - 8714 p T^{7} + 3918 p^{2} T^{8} - 582 p^{3} T^{9} + 121 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - 14 T + 172 T^{2} - 1660 T^{3} + 13659 T^{4} - 99542 T^{5} + 654984 T^{6} - 99542 p T^{7} + 13659 p^{2} T^{8} - 1660 p^{3} T^{9} + 172 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 2 T + 182 T^{2} - 292 T^{3} + 15743 T^{4} - 20162 T^{5} + 811820 T^{6} - 20162 p T^{7} + 15743 p^{2} T^{8} - 292 p^{3} T^{9} + 182 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 9 T + 217 T^{2} - 1338 T^{3} + 18751 T^{4} - 87029 T^{5} + 968158 T^{6} - 87029 p T^{7} + 18751 p^{2} T^{8} - 1338 p^{3} T^{9} + 217 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 11 T + 229 T^{2} - 2150 T^{3} + 24133 T^{4} - 185555 T^{5} + 1459746 T^{6} - 185555 p T^{7} + 24133 p^{2} T^{8} - 2150 p^{3} T^{9} + 229 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 9 T + 263 T^{2} + 2036 T^{3} + 31575 T^{4} + 199839 T^{5} + 2160582 T^{6} + 199839 p T^{7} + 31575 p^{2} T^{8} + 2036 p^{3} T^{9} + 263 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 6 T + 286 T^{2} - 1476 T^{3} + 37139 T^{4} - 157902 T^{5} + 2801380 T^{6} - 157902 p T^{7} + 37139 p^{2} T^{8} - 1476 p^{3} T^{9} + 286 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 10 T + 270 T^{2} - 1926 T^{3} + 33463 T^{4} - 197128 T^{5} + 2588164 T^{6} - 197128 p T^{7} + 33463 p^{2} T^{8} - 1926 p^{3} T^{9} + 270 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 13 T + 273 T^{2} - 2166 T^{3} + 29611 T^{4} - 171349 T^{5} + 2110378 T^{6} - 171349 p T^{7} + 29611 p^{2} T^{8} - 2166 p^{3} T^{9} + 273 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 11 T + 303 T^{2} + 2404 T^{3} + 39613 T^{4} + 243041 T^{5} + 3296750 T^{6} + 243041 p T^{7} + 39613 p^{2} T^{8} + 2404 p^{3} T^{9} + 303 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 2 T + 234 T^{2} - 462 T^{3} + 19023 T^{4} - 154724 T^{5} + 1061548 T^{6} - 154724 p T^{7} + 19023 p^{2} T^{8} - 462 p^{3} T^{9} + 234 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 9 T + 331 T^{2} - 3078 T^{3} + 52559 T^{4} - 453681 T^{5} + 5148328 T^{6} - 453681 p T^{7} + 52559 p^{2} T^{8} - 3078 p^{3} T^{9} + 331 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 13 T + 485 T^{2} + 4868 T^{3} + 98147 T^{4} + 765883 T^{5} + 10753262 T^{6} + 765883 p T^{7} + 98147 p^{2} T^{8} + 4868 p^{3} T^{9} + 485 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 28 T + 603 T^{2} - 7724 T^{3} + 87250 T^{4} - 722692 T^{5} + 7067375 T^{6} - 722692 p T^{7} + 87250 p^{2} T^{8} - 7724 p^{3} T^{9} + 603 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 18 T + 469 T^{2} - 4974 T^{3} + 76322 T^{4} - 573306 T^{5} + 7817845 T^{6} - 573306 p T^{7} + 76322 p^{2} T^{8} - 4974 p^{3} T^{9} + 469 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.02408396260376721507434968045, −3.67248805423295888760343561394, −3.51105392300270833741275840282, −3.37175325034676989195079251661, −3.35283878221232450565331861123, −3.21856729419215938496672678283, −3.16824870907997050890950239926, −3.03312040373856331294475606641, −2.75011589950354864298953576007, −2.60586058984247326606700965497, −2.56986617549885358724909368365, −2.54214131911235251958325223123, −2.39030255970044035103030389756, −2.12327078433668227729419004017, −1.89021664169751925816201500970, −1.80514220548319033502555812841, −1.77047401917849930825016814221, −1.75145710785062695925508594674, −1.56219896141925438824579436834, −1.00778053719785233372264492470, −0.920285835675193742640019273300, −0.905065777732089944088187940502, −0.854667038378689526615026605236, −0.67922070917768276605036691188, −0.40719904314379385638941589312,
0.40719904314379385638941589312, 0.67922070917768276605036691188, 0.854667038378689526615026605236, 0.905065777732089944088187940502, 0.920285835675193742640019273300, 1.00778053719785233372264492470, 1.56219896141925438824579436834, 1.75145710785062695925508594674, 1.77047401917849930825016814221, 1.80514220548319033502555812841, 1.89021664169751925816201500970, 2.12327078433668227729419004017, 2.39030255970044035103030389756, 2.54214131911235251958325223123, 2.56986617549885358724909368365, 2.60586058984247326606700965497, 2.75011589950354864298953576007, 3.03312040373856331294475606641, 3.16824870907997050890950239926, 3.21856729419215938496672678283, 3.35283878221232450565331861123, 3.37175325034676989195079251661, 3.51105392300270833741275840282, 3.67248805423295888760343561394, 4.02408396260376721507434968045
Plot not available for L-functions of degree greater than 10.
