| L(s) = 1 | − 32·4-s − 18·5-s − 162·7-s + 6·11-s + 640·16-s − 510·17-s − 12·19-s + 576·20-s + 396·23-s − 609·25-s + 5.18e3·28-s + 2.91e3·35-s − 8.65e3·43-s − 192·44-s − 3.21e3·47-s + 8.12e3·49-s − 108·55-s + 1.31e3·61-s − 1.02e4·64-s + 1.63e4·68-s + 2.33e4·73-s + 384·76-s − 972·77-s − 1.15e4·80-s + 1.04e4·83-s + 9.18e3·85-s − 1.26e4·92-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.719·5-s − 3.30·7-s + 0.0495·11-s + 5/2·16-s − 1.76·17-s − 0.0332·19-s + 1.43·20-s + 0.748·23-s − 0.974·25-s + 6.61·28-s + 2.38·35-s − 4.68·43-s − 0.0991·44-s − 1.45·47-s + 3.38·49-s − 0.0357·55-s + 0.353·61-s − 5/2·64-s + 3.52·68-s + 4.39·73-s + 0.0664·76-s − 0.163·77-s − 9/5·80-s + 1.51·83-s + 1.27·85-s − 1.49·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4235026784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4235026784\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 12 T + 303560 T^{2} - 1063956 p T^{3} + 6709482 p^{3} T^{4} - 1063956 p^{5} T^{5} + 303560 p^{8} T^{6} + 12 p^{12} T^{7} + p^{16} T^{8} \) |
| good | 5 | \( ( 1 + 9 T + 426 T^{2} + 10251 T^{3} + 716146 T^{4} + 10251 p^{4} T^{5} + 426 p^{8} T^{6} + 9 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 7 | \( ( 1 + 81 T + 5777 T^{2} + 49050 p T^{3} + 20657670 T^{4} + 49050 p^{5} T^{5} + 5777 p^{8} T^{6} + 81 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3 T + 20874 T^{2} - 203751 p T^{3} + 23528702 p T^{4} - 203751 p^{5} T^{5} + 20874 p^{8} T^{6} - 3 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 13 | \( 1 - 87902 T^{2} + 4031399833 T^{4} - 153863832997766 T^{6} + 5034444913352767444 T^{8} - 153863832997766 p^{8} T^{10} + 4031399833 p^{16} T^{12} - 87902 p^{24} T^{14} + p^{32} T^{16} \) |
| 17 | \( ( 1 + 15 p T + 266529 T^{2} + 48908322 T^{3} + 30973446334 T^{4} + 48908322 p^{4} T^{5} + 266529 p^{8} T^{6} + 15 p^{13} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( ( 1 - 198 T + 793305 T^{2} - 151845534 T^{3} + 305726793412 T^{4} - 151845534 p^{4} T^{5} + 793305 p^{8} T^{6} - 198 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 29 | \( 1 - 1884206 T^{2} + 2536130733097 T^{4} - 2411241746303389430 T^{6} + \)\(19\!\cdots\!76\)\( T^{8} - 2411241746303389430 p^{8} T^{10} + 2536130733097 p^{16} T^{12} - 1884206 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( 1 - 5185760 T^{2} + 13147692978364 T^{4} - 21074811809060168480 T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - 21074811809060168480 p^{8} T^{10} + 13147692978364 p^{16} T^{12} - 5185760 p^{24} T^{14} + p^{32} T^{16} \) |
| 37 | \( 1 - 8989760 T^{2} + 32058963480700 T^{4} - 61405971768293818304 T^{6} + \)\(98\!\cdots\!62\)\( T^{8} - 61405971768293818304 p^{8} T^{10} + 32058963480700 p^{16} T^{12} - 8989760 p^{24} T^{14} + p^{32} T^{16} \) |
| 41 | \( 1 - 15418400 T^{2} + 116131174644796 T^{4} - \)\(56\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!30\)\( T^{8} - \)\(56\!\cdots\!20\)\( p^{8} T^{10} + 116131174644796 p^{16} T^{12} - 15418400 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 4327 T + 17040418 T^{2} + 41562268261 T^{3} + 93546947175394 T^{4} + 41562268261 p^{4} T^{5} + 17040418 p^{8} T^{6} + 4327 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 1605 T + 15846978 T^{2} + 24977693547 T^{3} + 106935047396698 T^{4} + 24977693547 p^{4} T^{5} + 15846978 p^{8} T^{6} + 1605 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 53 | \( 1 - 26276750 T^{2} + 354841909979401 T^{4} - \)\(39\!\cdots\!70\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{8} - \)\(39\!\cdots\!70\)\( p^{8} T^{10} + 354841909979401 p^{16} T^{12} - 26276750 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 70986542 T^{2} + 2266647860250313 T^{4} - \)\(44\!\cdots\!70\)\( T^{6} + \)\(62\!\cdots\!16\)\( T^{8} - \)\(44\!\cdots\!70\)\( p^{8} T^{10} + 2266647860250313 p^{16} T^{12} - 70986542 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 657 T + 13651298 T^{2} + 36749564541 T^{3} + 190575679355970 T^{4} + 36749564541 p^{4} T^{5} + 13651298 p^{8} T^{6} - 657 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( 1 - 11587598 T^{2} + 1243984422721705 T^{4} - \)\(11\!\cdots\!74\)\( T^{6} + \)\(71\!\cdots\!88\)\( T^{8} - \)\(11\!\cdots\!74\)\( p^{8} T^{10} + 1243984422721705 p^{16} T^{12} - 11587598 p^{24} T^{14} + p^{32} T^{16} \) |
| 71 | \( 1 - 27489248 T^{2} + 2078050971545020 T^{4} - \)\(45\!\cdots\!44\)\( T^{6} + \)\(19\!\cdots\!38\)\( T^{8} - \)\(45\!\cdots\!44\)\( p^{8} T^{10} + 2078050971545020 p^{16} T^{12} - 27489248 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 11699 T + 88585093 T^{2} - 366782768102 T^{3} + 1847388837013414 T^{4} - 366782768102 p^{4} T^{5} + 88585093 p^{8} T^{6} - 11699 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( 1 - 198341024 T^{2} + 17865606744931900 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(43\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!64\)\( p^{8} T^{10} + 17865606744931900 p^{16} T^{12} - 198341024 p^{24} T^{14} + p^{32} T^{16} \) |
| 83 | \( ( 1 - 5220 T + 132074424 T^{2} - 739113578028 T^{3} + 8089121462686894 T^{4} - 739113578028 p^{4} T^{5} + 132074424 p^{8} T^{6} - 5220 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 89 | \( 1 - 219510728 T^{2} + 29593428331358620 T^{4} - \)\(27\!\cdots\!44\)\( T^{6} + \)\(20\!\cdots\!18\)\( T^{8} - \)\(27\!\cdots\!44\)\( p^{8} T^{10} + 29593428331358620 p^{16} T^{12} - 219510728 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( 1 - 272222816 T^{2} + 39430845297013180 T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(48\!\cdots\!14\)\( T^{8} - \)\(47\!\cdots\!04\)\( p^{8} T^{10} + 39430845297013180 p^{16} T^{12} - 272222816 p^{24} T^{14} + p^{32} T^{16} \) |
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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
−4.35446486310095731372967005768, −4.33863789148660415200738568156, −3.98671778366609736595454615714, −3.93241709228452868257124826810, −3.73901190451142807992977957460, −3.60321735081259257813228492503, −3.58967585979512576948341888573, −3.36485011581419257762312239246, −3.30439067428263954115682658978, −3.16238184890857202105181457884, −2.88839049028144617443152947074, −2.85581039021925159902575661543, −2.53812175616650603826502296734, −2.31717606750502836197162347864, −2.26761502979369915566635992157, −1.97945500767073243381566866565, −1.69326940337385670972221626952, −1.48072957676326213467968520091, −1.33869374863725754631994652274, −1.14457683803292809307398670029, −0.824933295848586916747602437065, −0.34128637855499671756155367090, −0.33361031978488663013352474787, −0.28898870728559711450147541191, −0.24438477800554772552297965093,
0.24438477800554772552297965093, 0.28898870728559711450147541191, 0.33361031978488663013352474787, 0.34128637855499671756155367090, 0.824933295848586916747602437065, 1.14457683803292809307398670029, 1.33869374863725754631994652274, 1.48072957676326213467968520091, 1.69326940337385670972221626952, 1.97945500767073243381566866565, 2.26761502979369915566635992157, 2.31717606750502836197162347864, 2.53812175616650603826502296734, 2.85581039021925159902575661543, 2.88839049028144617443152947074, 3.16238184890857202105181457884, 3.30439067428263954115682658978, 3.36485011581419257762312239246, 3.58967585979512576948341888573, 3.60321735081259257813228492503, 3.73901190451142807992977957460, 3.93241709228452868257124826810, 3.98671778366609736595454615714, 4.33863789148660415200738568156, 4.35446486310095731372967005768
Plot not available for L-functions of degree greater than 10.
