| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 8·6-s − 2·7-s + 4·8-s + 6·9-s − 16·12-s − 4·13-s − 4·14-s + 6·16-s + 2·17-s + 12·18-s + 12·19-s + 8·21-s + 10·23-s − 16·24-s − 8·26-s − 8·28-s − 12·29-s + 8·31-s + 4·32-s + 4·34-s + 24·36-s + 24·37-s + 24·38-s + 16·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 2·4-s − 3.26·6-s − 0.755·7-s + 1.41·8-s + 2·9-s − 4.61·12-s − 1.10·13-s − 1.06·14-s + 3/2·16-s + 0.485·17-s + 2.82·18-s + 2.75·19-s + 1.74·21-s + 2.08·23-s − 3.26·24-s − 1.56·26-s − 1.51·28-s − 2.22·29-s + 1.43·31-s + 0.707·32-s + 0.685·34-s + 4·36-s + 3.94·37-s + 3.89·38-s + 2.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.498041979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.498041979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2 T - 8 T^{2} - 2 p T^{3} + 41 T^{4} - 2 p^{2} T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( 1 - p T + p^{2} T^{3} - 3 p T^{4} + p^{2} T^{5} + p^{2} T^{6} - 3 p^{2} T^{7} + 3 p^{2} T^{8} - 3 p^{3} T^{9} + p^{4} T^{10} + p^{5} T^{11} - 3 p^{5} T^{12} + p^{7} T^{13} - p^{8} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 2968 p T^{9} - 2872 p^{2} T^{10} + 518 p^{3} T^{11} + 316 p^{4} T^{12} - 28 p^{5} T^{13} - 26 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 2 T + 24 T^{2} + 42 T^{3} + 413 T^{4} + 42 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 2 T - 36 T^{2} + 80 T^{3} + 524 T^{4} - 896 T^{5} - 9840 T^{6} + 2574 T^{7} + 223639 T^{8} + 2574 p T^{9} - 9840 p^{2} T^{10} - 896 p^{3} T^{11} + 524 p^{4} T^{12} + 80 p^{5} T^{13} - 36 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 12 T + 30 T^{2} + 80 T^{3} + 689 T^{4} - 9436 T^{5} + 28846 T^{6} - 59944 T^{7} + 291332 T^{8} - 59944 p T^{9} + 28846 p^{2} T^{10} - 9436 p^{3} T^{11} + 689 p^{4} T^{12} + 80 p^{5} T^{13} + 30 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 10 T + 40 T^{2} - 392 T^{3} + 2364 T^{4} - 4712 T^{5} + 38616 T^{6} - 172414 T^{7} + 38807 T^{8} - 172414 p T^{9} + 38616 p^{2} T^{10} - 4712 p^{3} T^{11} + 2364 p^{4} T^{12} - 392 p^{5} T^{13} + 40 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 332 p T^{5} + 78 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 60416 p T^{9} - 157694 p^{2} T^{10} - 9272 p^{3} T^{11} + 3285 p^{4} T^{12} + 352 p^{5} T^{13} - 54 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 24 T + 224 T^{2} - 1676 T^{3} + 18423 T^{4} - 154738 T^{5} + 24260 p T^{6} - 6581218 T^{7} + 49111756 T^{8} - 6581218 p T^{9} + 24260 p^{3} T^{10} - 154738 p^{3} T^{11} + 18423 p^{4} T^{12} - 1676 p^{5} T^{13} + 224 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 346 p T^{5} + 114 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 140 T^{2} + 878 T^{3} + 8293 T^{4} + 878 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 10 T - 104 T^{2} - 652 T^{3} + 13518 T^{4} + 45086 T^{5} - 19272 p T^{6} - 563558 T^{7} + 53075567 T^{8} - 563558 p T^{9} - 19272 p^{3} T^{10} + 45086 p^{3} T^{11} + 13518 p^{4} T^{12} - 652 p^{5} T^{13} - 104 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 20 T + 68 T^{2} + 136 T^{3} + 12794 T^{4} - 117580 T^{5} - 98000 T^{6} - 2920764 T^{7} + 73537971 T^{8} - 2920764 p T^{9} - 98000 p^{2} T^{10} - 117580 p^{3} T^{11} + 12794 p^{4} T^{12} + 136 p^{5} T^{13} + 68 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 2 T - 226 T^{2} - 4 p T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 407278 p T^{9} - 2961940 p^{2} T^{10} + 19820 p^{3} T^{11} + 32016 p^{4} T^{12} - 4 p^{6} T^{13} - 226 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 798752 p T^{9} - 247632 p^{2} T^{10} + 13396 p^{3} T^{11} + 3998 p^{4} T^{12} + 376 p^{5} T^{13} - 80 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 6 T - 160 T^{2} + 148 T^{3} + 18063 T^{4} + 22466 T^{5} - 1278868 T^{6} - 541096 T^{7} + 68029204 T^{8} - 541096 p T^{9} - 1278868 p^{2} T^{10} + 22466 p^{3} T^{11} + 18063 p^{4} T^{12} + 148 p^{5} T^{13} - 160 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 1648 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 12 T - 84 T^{2} + 932 T^{3} + 8735 T^{4} - 42250 T^{5} - 655952 T^{6} + 2845850 T^{7} + 12256412 T^{8} + 2845850 p T^{9} - 655952 p^{2} T^{10} - 42250 p^{3} T^{11} + 8735 p^{4} T^{12} + 932 p^{5} T^{13} - 84 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 8 T - 98 T^{2} + 1128 T^{3} - 3647 T^{4} + 31028 T^{5} - 546546 T^{6} - 5486636 T^{7} + 152161556 T^{8} - 5486636 p T^{9} - 546546 p^{2} T^{10} + 31028 p^{3} T^{11} - 3647 p^{4} T^{12} + 1128 p^{5} T^{13} - 98 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 6 T + 276 T^{2} - 1256 T^{3} + 32400 T^{4} - 1256 p T^{5} + 276 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 32888 p T^{7} - 1604897 p T^{8} - 32888 p^{2} T^{9} + 1051584 p^{2} T^{10} + 89290 p^{3} T^{11} + 4584 p^{4} T^{12} - 1004 p^{5} T^{13} - 98 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 2 T + 380 T^{2} - 566 T^{3} + 54898 T^{4} - 566 p T^{5} + 380 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.96350490576413042907144839753, −4.88565208534300547621010569460, −4.46076560003190803785708728323, −4.43233789924703346092850190807, −4.19205879129927773356422991314, −4.06115425477116614411899827918, −3.86505843824853522230639303021, −3.77311040282285218532302401026, −3.72632806228961611230493363588, −3.62945550904974858246322930363, −3.26036272607782179356385401758, −3.10394197767145924216746535588, −2.90558072246960138254027469246, −2.77642045728775285617246311716, −2.69842669180661025947812556858, −2.61200751434101947160734884376, −2.50591571573454311537310015952, −2.42255962811766759703454495267, −1.75562157789578956132155190117, −1.61543986297855607711894565233, −1.46268898536952160928111964946, −1.26553505576720249603792869300, −0.815894327006168871616067514275, −0.62483360866398971195416343925, −0.52045807961786981756271895188,
0.52045807961786981756271895188, 0.62483360866398971195416343925, 0.815894327006168871616067514275, 1.26553505576720249603792869300, 1.46268898536952160928111964946, 1.61543986297855607711894565233, 1.75562157789578956132155190117, 2.42255962811766759703454495267, 2.50591571573454311537310015952, 2.61200751434101947160734884376, 2.69842669180661025947812556858, 2.77642045728775285617246311716, 2.90558072246960138254027469246, 3.10394197767145924216746535588, 3.26036272607782179356385401758, 3.62945550904974858246322930363, 3.72632806228961611230493363588, 3.77311040282285218532302401026, 3.86505843824853522230639303021, 4.06115425477116614411899827918, 4.19205879129927773356422991314, 4.43233789924703346092850190807, 4.46076560003190803785708728323, 4.88565208534300547621010569460, 4.96350490576413042907144839753
Plot not available for L-functions of degree greater than 10.
