| L(s) = 1 | − 6·3-s − 4-s + 10·7-s + 23·9-s + 6·12-s + 10·13-s − 16-s − 20·19-s − 60·21-s + 25-s − 70·27-s − 10·28-s − 20·31-s − 23·36-s − 6·37-s − 60·39-s + 6·48-s + 51·49-s − 10·52-s + 120·57-s + 10·61-s + 230·63-s − 5·64-s − 4·67-s − 6·75-s + 20·76-s − 50·79-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 1/2·4-s + 3.77·7-s + 23/3·9-s + 1.73·12-s + 2.77·13-s − 1/4·16-s − 4.58·19-s − 13.0·21-s + 1/5·25-s − 13.4·27-s − 1.88·28-s − 3.59·31-s − 3.83·36-s − 0.986·37-s − 9.60·39-s + 0.866·48-s + 51/7·49-s − 1.38·52-s + 15.8·57-s + 1.28·61-s + 28.9·63-s − 5/8·64-s − 0.488·67-s − 0.692·75-s + 2.29·76-s − 5.62·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3377720842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3377720842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 p T + 13 T^{2} + 10 T^{3} + T^{4} + 10 p T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T^{2} + p T^{4} + p^{3} T^{6} + 25 T^{8} + p^{5} T^{10} + p^{5} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - T^{2} + 51 T^{4} + 109 T^{6} + 1136 T^{8} + 109 p^{2} T^{10} + 51 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 5 T + 12 T^{2} + 5 T^{3} - 51 T^{4} + 5 p T^{5} + 12 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 5 T + 18 T^{2} + 5 T^{3} - 21 T^{4} + 5 p T^{5} + 18 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 2 p T^{2} + 267 T^{4} + 8548 T^{6} - 277795 T^{8} + 8548 p^{2} T^{10} + 267 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 10 T + 69 T^{2} + 410 T^{3} + 1911 T^{4} + 410 p T^{5} + 69 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 50 T^{2} + 1363 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 2 p T^{2} + 1523 T^{4} - 6056 T^{6} - 679595 T^{8} - 6056 p^{2} T^{10} + 1523 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 10 T + 9 T^{2} - 130 T^{3} - 409 T^{4} - 130 p T^{5} + 9 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 87 T^{2} + 4308 T^{4} - 137029 T^{6} + 5286975 T^{8} - 137029 p^{2} T^{10} + 4308 p^{4} T^{12} - 87 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 122 T^{2} + 6919 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 15 T^{2} - 969 T^{4} - 93835 T^{6} + 1517976 T^{8} - 93835 p^{2} T^{10} - 969 p^{4} T^{12} + 15 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 70 T^{2} + 6051 T^{4} + 452660 T^{6} + 19250861 T^{8} + 452660 p^{2} T^{10} + 6051 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 122 T^{2} + 9663 T^{4} + 604444 T^{6} + 40851365 T^{8} + 604444 p^{2} T^{10} + 9663 p^{4} T^{12} + 122 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 5 T + p T^{2} + 5 p T^{3} + 796 T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 - 13 T^{2} + 5103 T^{4} - 100031 T^{6} + 31068680 T^{8} - 100031 p^{2} T^{10} + 5103 p^{4} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + p T^{2} + 360 T^{3} + 2449 T^{4} + 360 p T^{5} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 25 T + 304 T^{2} + 2435 T^{3} + 19501 T^{4} + 2435 p T^{5} + 304 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 149 T^{2} + 16452 T^{4} + 1844407 T^{6} + 200466815 T^{8} + 1844407 p^{2} T^{10} + 16452 p^{4} T^{12} + 149 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 266 T^{2} + 31531 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 3 T - 63 T^{2} + 835 T^{3} + 4236 T^{4} + 835 p T^{5} - 63 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−5.13161853290926275920356755934, −5.06661595328533012418479376994, −5.03929855110786771276721600455, −4.55789483018797076670353202985, −4.40805636460600303528102390698, −4.37329514535236683122641015280, −4.28784193248554387997154953220, −4.16123765551905636689403451355, −4.09334968907129957071984549337, −4.07352471200031155382981886419, −3.77060262560312772726681253761, −3.74391992654844642399222668308, −3.31887690127537012266958993132, −3.14493119866640208778400768995, −2.88368423328129177694388737918, −2.55757890794141277091365585383, −2.15779757305991594503529318329, −1.92780243362856542071915554372, −1.86375941074100668139664337168, −1.82145134085936736166164170595, −1.54792979158296001989880057030, −1.45037382437452631066300191552, −1.20047618005807289813147778038, −0.73383767677825996304746693894, −0.19705575038837811389160996860,
0.19705575038837811389160996860, 0.73383767677825996304746693894, 1.20047618005807289813147778038, 1.45037382437452631066300191552, 1.54792979158296001989880057030, 1.82145134085936736166164170595, 1.86375941074100668139664337168, 1.92780243362856542071915554372, 2.15779757305991594503529318329, 2.55757890794141277091365585383, 2.88368423328129177694388737918, 3.14493119866640208778400768995, 3.31887690127537012266958993132, 3.74391992654844642399222668308, 3.77060262560312772726681253761, 4.07352471200031155382981886419, 4.09334968907129957071984549337, 4.16123765551905636689403451355, 4.28784193248554387997154953220, 4.37329514535236683122641015280, 4.40805636460600303528102390698, 4.55789483018797076670353202985, 5.03929855110786771276721600455, 5.06661595328533012418479376994, 5.13161853290926275920356755934
Plot not available for L-functions of degree greater than 10.
