L(s) = 1 | − 4·3-s + 4-s + 2·5-s + 6·9-s + 11-s − 4·12-s − 8·15-s + 16-s + 2·20-s − 9·23-s + 3·25-s − 4·27-s − 31-s − 4·33-s + 6·36-s − 37-s + 44-s + 12·45-s − 47-s − 4·48-s + 49-s + 2·53-s + 2·55-s + 2·59-s − 8·60-s + 67-s + 36·69-s + ⋯ |
L(s) = 1 | − 4·3-s + 4-s + 2·5-s + 6·9-s + 11-s − 4·12-s − 8·15-s + 16-s + 2·20-s − 9·23-s + 3·25-s − 4·27-s − 31-s − 4·33-s + 6·36-s − 37-s + 44-s + 12·45-s − 47-s − 4·48-s + 49-s + 2·53-s + 2·55-s + 2·59-s − 8·60-s + 67-s + 36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03795126939\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03795126939\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
| 67 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
good | 2 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 7 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 23 | \( ( 1 + T + T^{2} )^{10}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 29 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 37 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 47 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 71 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 79 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.56707178261188949432709710975, −2.39538959573792573528500386974, −2.38212142008290081637571970405, −2.37274733608426455280472503810, −2.26102553418900920314007930962, −2.26080977152802948409920279808, −2.22626740501381137516715091970, −2.21291325523249749897468364674, −2.07923674563557950027010530554, −2.02161964333161422215167784346, −2.00652595547490685114943758013, −1.88235187336754045450457269563, −1.86711090748898397290393723698, −1.71239564710703301069093858980, −1.69720469005374246181726391599, −1.69652622325820957446652078205, −1.41648548041846735607893900870, −1.33170745976490825633934786403, −1.31585374269621423602568668769, −1.29279331669749068583501158313, −1.17090733415714998701305698336, −0.938114145020851767251685537413, −0.873462383892445419247479014050, −0.63608141169059796707986662926, −0.52393109863700943502525664586,
0.52393109863700943502525664586, 0.63608141169059796707986662926, 0.873462383892445419247479014050, 0.938114145020851767251685537413, 1.17090733415714998701305698336, 1.29279331669749068583501158313, 1.31585374269621423602568668769, 1.33170745976490825633934786403, 1.41648548041846735607893900870, 1.69652622325820957446652078205, 1.69720469005374246181726391599, 1.71239564710703301069093858980, 1.86711090748898397290393723698, 1.88235187336754045450457269563, 2.00652595547490685114943758013, 2.02161964333161422215167784346, 2.07923674563557950027010530554, 2.21291325523249749897468364674, 2.22626740501381137516715091970, 2.26080977152802948409920279808, 2.26102553418900920314007930962, 2.37274733608426455280472503810, 2.38212142008290081637571970405, 2.39538959573792573528500386974, 2.56707178261188949432709710975
Plot not available for L-functions of degree greater than 10.