| L(s) = 1 | + 3-s + 28·7-s − 21·9-s + 36·11-s + 94·13-s − 53·17-s − 91·19-s + 28·21-s − 63·23-s + 118·27-s − 103·29-s + 492·31-s + 36·33-s + 387·37-s + 94·39-s − 475·41-s − 213·43-s − 52·47-s + 490·49-s − 53·51-s − 378·53-s − 91·57-s − 548·59-s − 704·61-s − 588·63-s − 248·67-s − 63·69-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 1.51·7-s − 7/9·9-s + 0.986·11-s + 2.00·13-s − 0.756·17-s − 1.09·19-s + 0.290·21-s − 0.571·23-s + 0.841·27-s − 0.659·29-s + 2.85·31-s + 0.189·33-s + 1.71·37-s + 0.385·39-s − 1.80·41-s − 0.755·43-s − 0.161·47-s + 10/7·49-s − 0.145·51-s − 0.979·53-s − 0.211·57-s − 1.20·59-s − 1.47·61-s − 1.17·63-s − 0.452·67-s − 0.109·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.38283794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.38283794\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 22 T^{2} - 161 T^{3} + 826 T^{4} - 161 p^{3} T^{5} + 22 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 36 T + 3246 T^{2} - 105704 T^{3} + 6275815 T^{4} - 105704 p^{3} T^{5} + 3246 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 94 T + 4192 T^{2} - 155762 T^{3} + 6324830 T^{4} - 155762 p^{3} T^{5} + 4192 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 53 T + 16220 T^{2} + 593663 T^{3} + 6413142 p T^{4} + 593663 p^{3} T^{5} + 16220 p^{6} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 91 T + 26430 T^{2} + 1755535 T^{3} + 269614922 T^{4} + 1755535 p^{3} T^{5} + 26430 p^{6} T^{6} + 91 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 63 T + 41763 T^{2} + 2051708 T^{3} + 735601576 T^{4} + 2051708 p^{3} T^{5} + 41763 p^{6} T^{6} + 63 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 103 T - 12915 T^{2} + 1943726 T^{3} + 1081092170 T^{4} + 1943726 p^{3} T^{5} - 12915 p^{6} T^{6} + 103 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 492 T + 113576 T^{2} - 18107732 T^{3} + 2932839374 T^{4} - 18107732 p^{3} T^{5} + 113576 p^{6} T^{6} - 492 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 387 T + 201103 T^{2} - 55567686 T^{3} + 15218614268 T^{4} - 55567686 p^{3} T^{5} + 201103 p^{6} T^{6} - 387 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 475 T + 240478 T^{2} + 59971201 T^{3} + 19618769394 T^{4} + 59971201 p^{3} T^{5} + 240478 p^{6} T^{6} + 475 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 213 T + 135019 T^{2} + 18207912 T^{3} + 13522925048 T^{4} + 18207912 p^{3} T^{5} + 135019 p^{6} T^{6} + 213 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 52 T + 139884 T^{2} - 53490476 T^{3} + 3799917334 T^{4} - 53490476 p^{3} T^{5} + 139884 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 378 T + 345656 T^{2} + 90143982 T^{3} + 64834489406 T^{4} + 90143982 p^{3} T^{5} + 345656 p^{6} T^{6} + 378 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 548 T + 883368 T^{2} + 5660100 p T^{3} + 278433047806 T^{4} + 5660100 p^{4} T^{5} + 883368 p^{6} T^{6} + 548 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 704 T + 239744 T^{2} - 134132472 T^{3} - 95245640930 T^{4} - 134132472 p^{3} T^{5} + 239744 p^{6} T^{6} + 704 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 248 T + 813042 T^{2} + 123084488 T^{3} + 323899445819 T^{4} + 123084488 p^{3} T^{5} + 813042 p^{6} T^{6} + 248 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 635 T + 1383137 T^{2} + 640675164 T^{3} + 736363936782 T^{4} + 640675164 p^{3} T^{5} + 1383137 p^{6} T^{6} + 635 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 959 T + 1401710 T^{2} - 900008701 T^{3} + 767880046722 T^{4} - 900008701 p^{3} T^{5} + 1401710 p^{6} T^{6} - 959 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 151 T + 1139005 T^{2} + 51686620 T^{3} + 632152934042 T^{4} + 51686620 p^{3} T^{5} + 1139005 p^{6} T^{6} - 151 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1969 T + 3508668 T^{2} - 3610124949 T^{3} + 3359590271566 T^{4} - 3610124949 p^{3} T^{5} + 3508668 p^{6} T^{6} - 1969 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 965 T + 1935460 T^{2} - 1800557379 T^{3} + 1722188867198 T^{4} - 1800557379 p^{3} T^{5} + 1935460 p^{6} T^{6} - 965 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1006 T + 3422824 T^{2} - 2463556314 T^{3} + 4534227353694 T^{4} - 2463556314 p^{3} T^{5} + 3422824 p^{6} T^{6} - 1006 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.53094920154293374152238968538, −6.22134621340801007331469284823, −5.95582439887854684171125214985, −5.94219622292811817295013665516, −5.71724479032454339563208031090, −5.14770463899026673037558422132, −5.12121610421802162048251506492, −4.90212844681317725028326412486, −4.53555079112756259336746755959, −4.40815868878173574570699236030, −4.20300217446670576453722552660, −4.07211239935196249800983241425, −3.84788937440934238632988667664, −3.21547668368904200722186764100, −3.20511040907617267805231612482, −2.98960615147443353904798125088, −2.84466157994126641631963676554, −2.24341657814142946773669041663, −1.90152853900704473336291270258, −1.85403829675206187510985353614, −1.61302760317761018669073607820, −1.28650674810752690146703150353, −0.72636508937592186523907670316, −0.71777140620769852622910938964, −0.37689160923227699242107769485,
0.37689160923227699242107769485, 0.71777140620769852622910938964, 0.72636508937592186523907670316, 1.28650674810752690146703150353, 1.61302760317761018669073607820, 1.85403829675206187510985353614, 1.90152853900704473336291270258, 2.24341657814142946773669041663, 2.84466157994126641631963676554, 2.98960615147443353904798125088, 3.20511040907617267805231612482, 3.21547668368904200722186764100, 3.84788937440934238632988667664, 4.07211239935196249800983241425, 4.20300217446670576453722552660, 4.40815868878173574570699236030, 4.53555079112756259336746755959, 4.90212844681317725028326412486, 5.12121610421802162048251506492, 5.14770463899026673037558422132, 5.71724479032454339563208031090, 5.94219622292811817295013665516, 5.95582439887854684171125214985, 6.22134621340801007331469284823, 6.53094920154293374152238968538
