| L(s) = 1 | + 7·4-s + 4·5-s + 10·9-s + 18·16-s + 12·19-s + 28·20-s + 4·25-s + 24·29-s + 14·31-s + 70·36-s − 34·41-s + 40·45-s + 43·49-s + 6·59-s − 20·61-s + 2·64-s − 42·71-s + 84·76-s + 16·79-s + 72·80-s + 38·81-s + 12·89-s + 48·95-s + 28·100-s − 58·101-s − 10·109-s + 168·116-s + ⋯ |
| L(s) = 1 | + 7/2·4-s + 1.78·5-s + 10/3·9-s + 9/2·16-s + 2.75·19-s + 6.26·20-s + 4/5·25-s + 4.45·29-s + 2.51·31-s + 35/3·36-s − 5.30·41-s + 5.96·45-s + 43/7·49-s + 0.781·59-s − 2.56·61-s + 1/4·64-s − 4.98·71-s + 9.63·76-s + 1.80·79-s + 8.04·80-s + 38/9·81-s + 1.27·89-s + 4.92·95-s + 14/5·100-s − 5.77·101-s − 0.957·109-s + 15.5·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(54.55739892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(54.55739892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 7 T^{2} + 31 T^{4} - 93 T^{6} + 215 T^{8} - 93 p^{2} T^{10} + 31 p^{4} T^{12} - 7 p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 - 10 T^{2} + 62 T^{4} - 275 T^{6} + 923 T^{8} - 275 p^{2} T^{10} + 62 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 43 T^{2} + 125 p T^{4} - 10978 T^{6} + 92649 T^{8} - 10978 p^{2} T^{10} + 125 p^{5} T^{12} - 43 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 59 T^{2} + 1897 T^{4} - 40182 T^{6} + 612545 T^{8} - 40182 p^{2} T^{10} + 1897 p^{4} T^{12} - 59 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 55 T^{2} + 1672 T^{4} - 33725 T^{6} + 594553 T^{8} - 33725 p^{2} T^{10} + 1672 p^{4} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 6 T + 72 T^{2} - 303 T^{3} + 2025 T^{4} - 303 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 85 T^{2} + 3382 T^{4} - 84765 T^{6} + 1856153 T^{8} - 84765 p^{2} T^{10} + 3382 p^{4} T^{12} - 85 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 12 T + 164 T^{2} - 1111 T^{3} + 7841 T^{4} - 1111 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 7 T + 129 T^{2} - 650 T^{3} + 6075 T^{4} - 650 p T^{5} + 129 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 195 T^{2} + 18732 T^{4} - 1164885 T^{6} + 50923593 T^{8} - 1164885 p^{2} T^{10} + 18732 p^{4} T^{12} - 195 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 17 T + 212 T^{2} + 1653 T^{3} + 12053 T^{4} + 1653 p T^{5} + 212 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 171 T^{2} + 15457 T^{4} - 1005398 T^{6} + 49870565 T^{8} - 1005398 p^{2} T^{10} + 15457 p^{4} T^{12} - 171 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 185 T^{2} + 15341 T^{4} - 777020 T^{6} + 34162701 T^{8} - 777020 p^{2} T^{10} + 15341 p^{4} T^{12} - 185 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 175 T^{2} + 14707 T^{4} - 910820 T^{6} + 50738713 T^{8} - 910820 p^{2} T^{10} + 14707 p^{4} T^{12} - 175 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 3 T + 161 T^{2} - 502 T^{3} + 12035 T^{4} - 502 p T^{5} + 161 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 10 T + 3 p T^{2} + 1820 T^{3} + 15093 T^{4} + 1820 p T^{5} + 3 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 387 T^{2} + 71530 T^{4} - 8300957 T^{6} + 662638709 T^{8} - 8300957 p^{2} T^{10} + 71530 p^{4} T^{12} - 387 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 21 T + 374 T^{2} + 4041 T^{3} + 40515 T^{4} + 4041 p T^{5} + 374 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 287 T^{2} + 44630 T^{4} - 4819437 T^{6} + 395749629 T^{8} - 4819437 p^{2} T^{10} + 44630 p^{4} T^{12} - 287 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 8 T + 264 T^{2} - 1519 T^{3} + 28911 T^{4} - 1519 p T^{5} + 264 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 553 T^{2} + 141545 T^{4} - 21804528 T^{6} + 2207036949 T^{8} - 21804528 p^{2} T^{10} + 141545 p^{4} T^{12} - 553 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 6 T + 228 T^{2} - 1116 T^{3} + 26613 T^{4} - 1116 p T^{5} + 228 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 614 T^{2} + 174756 T^{4} - 30358060 T^{6} + 3544770165 T^{8} - 30358060 p^{2} T^{10} + 174756 p^{4} T^{12} - 614 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.55420821037672187995385505183, −4.48928111412439945334710620241, −4.47771390923837335977357044692, −4.46533799155870648294543250088, −4.08451024368023975809124143219, −3.94637866689386181616977940189, −3.75265556654776585212126995640, −3.59802657289447257250977543591, −3.57963331110942288720055787348, −3.14405134522959540898604341811, −3.01404531256783976179737767224, −2.88616389014417602352742694798, −2.79581971450744370846750751402, −2.61816435366610464825403701138, −2.50556973495352807517076776072, −2.50101592128793138426813582401, −2.43200048439432592379321563653, −1.84607051580969633546519366298, −1.73494623839693811560801598653, −1.60472329025461557159818716344, −1.55776196138830821402920479356, −1.38396351227975638019251832274, −1.28775756720930651115278415518, −0.918561727167837428096114961716, −0.69432690020900580355535071714,
0.69432690020900580355535071714, 0.918561727167837428096114961716, 1.28775756720930651115278415518, 1.38396351227975638019251832274, 1.55776196138830821402920479356, 1.60472329025461557159818716344, 1.73494623839693811560801598653, 1.84607051580969633546519366298, 2.43200048439432592379321563653, 2.50101592128793138426813582401, 2.50556973495352807517076776072, 2.61816435366610464825403701138, 2.79581971450744370846750751402, 2.88616389014417602352742694798, 3.01404531256783976179737767224, 3.14405134522959540898604341811, 3.57963331110942288720055787348, 3.59802657289447257250977543591, 3.75265556654776585212126995640, 3.94637866689386181616977940189, 4.08451024368023975809124143219, 4.46533799155870648294543250088, 4.47771390923837335977357044692, 4.48928111412439945334710620241, 4.55420821037672187995385505183
Plot not available for L-functions of degree greater than 10.
