| L(s) = 1 | − 26·7-s + 10·13-s + 562·19-s + 2.14e3·25-s − 374·31-s + 16·37-s + 136·43-s − 8.93e3·49-s + 3.87e3·61-s − 308·67-s − 7.80e3·73-s + 4.39e3·79-s − 260·91-s − 1.45e4·97-s − 1.75e4·103-s − 7.22e3·109-s + 6.73e4·121-s + 127-s + 131-s − 1.46e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.530·7-s + 0.0591·13-s + 1.55·19-s + 3.43·25-s − 0.389·31-s + 0.0116·37-s + 0.0735·43-s − 3.72·49-s + 1.04·61-s − 0.0686·67-s − 1.46·73-s + 0.703·79-s − 0.0313·91-s − 1.54·97-s − 1.65·103-s − 0.607·109-s + 4.60·121-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.826·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.966234251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.966234251\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2147 T^{2} + 2477509 T^{4} - 2147976218 T^{6} + 1511855308306 T^{8} - 2147976218 p^{8} T^{10} + 2477509 p^{16} T^{12} - 2147 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 + 13 T + 4723 T^{2} + 93076 T^{3} + 12134350 T^{4} + 93076 p^{4} T^{5} + 4723 p^{8} T^{6} + 13 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 - 67376 T^{2} + 2511150250 T^{4} - 60712318203824 T^{6} + 1049763807787064539 T^{8} - 60712318203824 p^{8} T^{10} + 2511150250 p^{16} T^{12} - 67376 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 5 T + 42079 T^{2} - 3738530 T^{3} + 1399773016 T^{4} - 3738530 p^{4} T^{5} + 42079 p^{8} T^{6} - 5 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - 6759733382202755 p^{8} T^{10} + 52320681154 p^{16} T^{12} - 288125 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 68843041 p^{4} T^{5} + 110170 p^{8} T^{6} - 281 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 103955 T^{2} + 204060092029 T^{4} - 14985633296690750 T^{6} + \)\(21\!\cdots\!46\)\( T^{8} - 14985633296690750 p^{8} T^{10} + 204060092029 p^{16} T^{12} - 103955 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 3708803 T^{2} + 6181658247973 T^{4} - 6420936064399980986 T^{6} + \)\(50\!\cdots\!30\)\( T^{8} - 6420936064399980986 p^{8} T^{10} + 6181658247973 p^{16} T^{12} - 3708803 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 187 T + 2550973 T^{2} + 331939744 T^{3} + 3082958795560 T^{4} + 331939744 p^{4} T^{5} + 2550973 p^{8} T^{6} + 187 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 1256575624 p^{4} T^{5} + 3611368 p^{8} T^{6} - 8 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 12649388 T^{2} + 1815770138378 p T^{4} - \)\(29\!\cdots\!76\)\( T^{6} + \)\(89\!\cdots\!55\)\( T^{8} - \)\(29\!\cdots\!76\)\( p^{8} T^{10} + 1815770138378 p^{17} T^{12} - 12649388 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 - 68 T + 12955228 T^{2} - 545371076 T^{3} + 65226299853025 T^{4} - 545371076 p^{4} T^{5} + 12955228 p^{8} T^{6} - 68 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 19981955 T^{2} + 227810016714829 T^{4} - \)\(17\!\cdots\!90\)\( T^{6} + \)\(45\!\cdots\!94\)\( p^{2} T^{8} - \)\(17\!\cdots\!90\)\( p^{8} T^{10} + 227810016714829 p^{16} T^{12} - 19981955 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - 84051566001475463360 p^{8} T^{10} + 115452291970684 p^{16} T^{12} - 5145920 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 31340696 T^{2} + 646808461063090 T^{4} - \)\(10\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!19\)\( T^{8} - \)\(10\!\cdots\!44\)\( p^{8} T^{10} + 646808461063090 p^{16} T^{12} - 31340696 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 1937 T + 14281603 T^{2} + 60991858006 T^{3} - 94182960283700 T^{4} + 60991858006 p^{4} T^{5} + 14281603 p^{8} T^{6} - 1937 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 154 T + 33859570 T^{2} - 10195927724 T^{3} + 608674748042419 T^{4} - 10195927724 p^{4} T^{5} + 33859570 p^{8} T^{6} + 154 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(11\!\cdots\!24\)\( p^{8} T^{10} + 3244147638477940 p^{16} T^{12} - 68871716 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 292589317519 p^{4} T^{5} + 59309470 p^{8} T^{6} + 3901 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2195 T + 92542939 T^{2} - 195388180760 T^{3} + 5006999286743146 T^{4} - 195388180760 p^{4} T^{5} + 92542939 p^{8} T^{6} - 2195 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 81188291 T^{2} + 8014262915096365 T^{4} - \)\(34\!\cdots\!54\)\( T^{6} + \)\(22\!\cdots\!34\)\( T^{8} - \)\(34\!\cdots\!54\)\( p^{8} T^{10} + 8014262915096365 p^{16} T^{12} - 81188291 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(48\!\cdots\!04\)\( p^{8} T^{10} + 46567064448316540 p^{16} T^{12} - 294759296 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 + 7282 T + 336254488 T^{2} + 1720884578884 T^{3} + 43500636050571385 T^{4} + 1720884578884 p^{4} T^{5} + 336254488 p^{8} T^{6} + 7282 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.54219553276456518447167748428, −4.30070538717015979364037951017, −4.21410915283703071753401570865, −4.02746800160578631756635176478, −3.67958903342371623337704307565, −3.66909700850142729990495161114, −3.45727610780841899292820432898, −3.44141129426084342059414891211, −3.14776544263943567013789503650, −3.03538464029662912114747700894, −2.91069942296527214158384822807, −2.74428653612920223552533495702, −2.61492192205415193896693174621, −2.48076503565933858365446580405, −2.16315295724526497017142371644, −1.96628063732239954741029966438, −1.72933031917667954452690632132, −1.44794937679274465918767643565, −1.36096554379665878650283967724, −1.27035578701988277326846755452, −1.05953631369501989428970409124, −0.71846396908406444568713997372, −0.57290106530001980208458702923, −0.37270013246158453099451678998, −0.14382187100554646409560518950,
0.14382187100554646409560518950, 0.37270013246158453099451678998, 0.57290106530001980208458702923, 0.71846396908406444568713997372, 1.05953631369501989428970409124, 1.27035578701988277326846755452, 1.36096554379665878650283967724, 1.44794937679274465918767643565, 1.72933031917667954452690632132, 1.96628063732239954741029966438, 2.16315295724526497017142371644, 2.48076503565933858365446580405, 2.61492192205415193896693174621, 2.74428653612920223552533495702, 2.91069942296527214158384822807, 3.03538464029662912114747700894, 3.14776544263943567013789503650, 3.44141129426084342059414891211, 3.45727610780841899292820432898, 3.66909700850142729990495161114, 3.67958903342371623337704307565, 4.02746800160578631756635176478, 4.21410915283703071753401570865, 4.30070538717015979364037951017, 4.54219553276456518447167748428
Plot not available for L-functions of degree greater than 10.
