| L(s) = 1 | + 4·2-s + 4·3-s + 6·4-s + 16·6-s − 2·7-s + 6·9-s + 2·11-s + 24·12-s − 8·14-s − 15·16-s − 10·17-s + 24·18-s − 4·19-s − 8·21-s + 8·22-s + 5·23-s + 8·25-s − 12·28-s − 6·29-s − 9·31-s − 24·32-s + 8·33-s − 40·34-s + 36·36-s + 14·37-s − 16·38-s + 8·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 3·4-s + 6.53·6-s − 0.755·7-s + 2·9-s + 0.603·11-s + 6.92·12-s − 2.13·14-s − 3.75·16-s − 2.42·17-s + 5.65·18-s − 0.917·19-s − 1.74·21-s + 1.70·22-s + 1.04·23-s + 8/5·25-s − 2.26·28-s − 1.11·29-s − 1.61·31-s − 4.24·32-s + 1.39·33-s − 6.85·34-s + 6·36-s + 2.30·37-s − 2.59·38-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.69494937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.69494937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( 1 + 2 T + 4 T^{2} - 10 T^{3} - 41 T^{4} - 10 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 6 T + 8 T^{2} + 6 p T^{3} - 129 T^{4} + 6 p^{2} T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )( 1 + 6 T + 4 p T^{2} + 54 T^{3} + 126 T^{4} + 54 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( 1 - 2 T - 28 T^{2} + 24 T^{3} + 467 T^{4} - 16 T^{5} - 6296 T^{6} - 6 p T^{7} + 70768 T^{8} - 6 p^{2} T^{9} - 6296 p^{2} T^{10} - 16 p^{3} T^{11} + 467 p^{4} T^{12} + 24 p^{5} T^{13} - 28 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 4 T^{2} - 48 T^{3} + 102 T^{4} - 48 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 10 T + 38 T^{2} + 108 T^{3} + 206 T^{4} - 1336 T^{5} - 8984 T^{6} - 35370 T^{7} - 163865 T^{8} - 35370 p T^{9} - 8984 p^{2} T^{10} - 1336 p^{3} T^{11} + 206 p^{4} T^{12} + 108 p^{5} T^{13} + 38 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 5 T - 7 T^{2} + 150 T^{3} - 1087 T^{4} + 4835 T^{5} - 2696 T^{6} - 124395 T^{7} + 1019134 T^{8} - 124395 p T^{9} - 2696 p^{2} T^{10} + 4835 p^{3} T^{11} - 1087 p^{4} T^{12} + 150 p^{5} T^{13} - 7 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 3 T + 35 T^{2} + 126 T^{3} + 1644 T^{4} + 126 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 9 T - 10 T^{2} + 99 T^{3} + 2389 T^{4} - 3816 T^{5} - 37270 T^{6} - 21222 T^{7} - 1039916 T^{8} - 21222 p T^{9} - 37270 p^{2} T^{10} - 3816 p^{3} T^{11} + 2389 p^{4} T^{12} + 99 p^{5} T^{13} - 10 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 14 T + 6 T^{2} + 404 T^{3} + 3338 T^{4} - 30972 T^{5} - 75992 T^{6} - 143810 T^{7} + 9727623 T^{8} - 143810 p T^{9} - 75992 p^{2} T^{10} - 30972 p^{3} T^{11} + 3338 p^{4} T^{12} + 404 p^{5} T^{13} + 6 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 4 T + 68 T^{2} + 14 T^{3} + 1870 T^{4} + 14 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 21 T + 286 T^{2} - 2685 T^{3} + 20244 T^{4} - 2685 p T^{5} + 286 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 7 T - 115 T^{2} - 570 T^{3} + 9629 T^{4} + 23693 T^{5} - 665834 T^{6} - 312465 T^{7} + 37853818 T^{8} - 312465 p T^{9} - 665834 p^{2} T^{10} + 23693 p^{3} T^{11} + 9629 p^{4} T^{12} - 570 p^{5} T^{13} - 115 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 7 T - 70 T^{2} + 555 T^{3} + 1331 T^{4} - 7760 T^{5} - 55964 T^{6} - 160488 T^{7} + 4947292 T^{8} - 160488 p T^{9} - 55964 p^{2} T^{10} - 7760 p^{3} T^{11} + 1331 p^{4} T^{12} + 555 p^{5} T^{13} - 70 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 7 T - 4 T^{2} - 801 T^{3} - 9787 T^{4} - 53284 T^{5} + 20182 T^{6} + 4041744 T^{7} + 50119324 T^{8} + 4041744 p T^{9} + 20182 p^{2} T^{10} - 53284 p^{3} T^{11} - 9787 p^{4} T^{12} - 801 p^{5} T^{13} - 4 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 23 T + 99 T^{2} + 232 T^{3} + 28103 T^{4} + 267459 T^{5} + 80314 T^{6} + 9522083 T^{7} + 207046356 T^{8} + 9522083 p T^{9} + 80314 p^{2} T^{10} + 267459 p^{3} T^{11} + 28103 p^{4} T^{12} + 232 p^{5} T^{13} + 99 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 6 T + 8 T^{2} - 228 T^{3} - 3146 T^{4} - 14094 T^{5} + 134216 T^{6} + 2358750 T^{7} + 2876335 T^{8} + 2358750 p T^{9} + 134216 p^{2} T^{10} - 14094 p^{3} T^{11} - 3146 p^{4} T^{12} - 228 p^{5} T^{13} + 8 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 2 T + 158 T^{2} - 380 T^{3} + 12382 T^{4} - 380 p T^{5} + 158 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 5 T - 81 T^{2} - 820 T^{3} + 8819 T^{4} + 57375 T^{5} + 870646 T^{6} - 6557885 T^{7} - 60486696 T^{8} - 6557885 p T^{9} + 870646 p^{2} T^{10} + 57375 p^{3} T^{11} + 8819 p^{4} T^{12} - 820 p^{5} T^{13} - 81 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 11 T - 198 T^{2} + 1403 T^{3} + 36653 T^{4} - 139464 T^{5} - 4271918 T^{6} + 2910262 T^{7} + 419752476 T^{8} + 2910262 p T^{9} - 4271918 p^{2} T^{10} - 139464 p^{3} T^{11} + 36653 p^{4} T^{12} + 1403 p^{5} T^{13} - 198 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 14 T + 272 T^{2} - 3200 T^{3} + 31417 T^{4} - 3200 p T^{5} + 272 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 10 T - 100 T^{2} + 576 T^{3} + 17966 T^{4} - 92512 T^{5} + 58864 T^{6} + 91722 p T^{7} - 61279061 T^{8} + 91722 p^{2} T^{9} + 58864 p^{2} T^{10} - 92512 p^{3} T^{11} + 17966 p^{4} T^{12} + 576 p^{5} T^{13} - 100 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 + T + 319 T^{2} + 232 T^{3} + 44116 T^{4} + 232 p T^{5} + 319 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.34695367096310123993979429012, −4.27657181975494744598299067629, −4.07649851679416914230629760012, −4.01681839210996036307627232106, −3.91437356586083488991105772987, −3.81887054726243178533368640560, −3.73073809940891471608590697294, −3.70616123147252652625851179420, −3.20757446524576624713459809430, −3.14230253370120729577936608194, −3.12544290773258517584730385880, −2.92650565585940772943296461021, −2.88288993892061144680541578947, −2.70564315008875863096012538597, −2.68118609878816025019581372597, −2.37704828708127324626432141570, −2.26770258926195477928654559980, −1.99926828918830973008556994687, −1.97606054308563600801443743768, −1.97032111655133572341905052556, −1.53698439465977553823294236674, −1.08349893738728165568500955602, −0.866527039500685051311370500646, −0.70217313964319743459085512356, −0.30720162304670363033147764612,
0.30720162304670363033147764612, 0.70217313964319743459085512356, 0.866527039500685051311370500646, 1.08349893738728165568500955602, 1.53698439465977553823294236674, 1.97032111655133572341905052556, 1.97606054308563600801443743768, 1.99926828918830973008556994687, 2.26770258926195477928654559980, 2.37704828708127324626432141570, 2.68118609878816025019581372597, 2.70564315008875863096012538597, 2.88288993892061144680541578947, 2.92650565585940772943296461021, 3.12544290773258517584730385880, 3.14230253370120729577936608194, 3.20757446524576624713459809430, 3.70616123147252652625851179420, 3.73073809940891471608590697294, 3.81887054726243178533368640560, 3.91437356586083488991105772987, 4.01681839210996036307627232106, 4.07649851679416914230629760012, 4.27657181975494744598299067629, 4.34695367096310123993979429012
Plot not available for L-functions of degree greater than 10.
