| L(s) = 1 | + 2·3-s + 4·5-s + 5·9-s − 2·11-s − 20·13-s + 8·15-s + 6·17-s + 4·23-s + 6·25-s + 2·27-s − 4·29-s − 12·31-s − 4·33-s − 40·39-s − 24·41-s − 16·43-s + 20·45-s − 2·47-s + 12·51-s + 4·53-s − 8·55-s + 8·59-s + 20·61-s − 80·65-s + 8·67-s + 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 5/3·9-s − 0.603·11-s − 5.54·13-s + 2.06·15-s + 1.45·17-s + 0.834·23-s + 6/5·25-s + 0.384·27-s − 0.742·29-s − 2.15·31-s − 0.696·33-s − 6.40·39-s − 3.74·41-s − 2.43·43-s + 2.98·45-s − 0.291·47-s + 1.68·51-s + 0.549·53-s − 1.07·55-s + 1.04·59-s + 2.56·61-s − 9.92·65-s + 0.977·67-s + 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994886789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994886789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T - T^{2} + 10 T^{3} - 7 p T^{4} + 8 p T^{5} - 104 T^{7} + 286 T^{8} - 104 p T^{9} + 8 p^{4} T^{11} - 7 p^{5} T^{12} + 10 p^{5} T^{13} - p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 2 T - 29 T^{2} - 26 T^{3} + 47 p T^{4} + 104 T^{5} - 6750 T^{6} - 668 T^{7} + 71362 T^{8} - 668 p T^{9} - 6750 p^{2} T^{10} + 104 p^{3} T^{11} + 47 p^{5} T^{12} - 26 p^{5} T^{13} - 29 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 10 T + 71 T^{2} + 2 p^{2} T^{3} + 1384 T^{4} + 2 p^{3} T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 6 T + 19 T^{2} + 22 T^{3} - 517 T^{4} + 3100 T^{5} - 32 T^{6} - 50028 T^{7} + 309702 T^{8} - 50028 p T^{9} - 32 p^{2} T^{10} + 3100 p^{3} T^{11} - 517 p^{4} T^{12} + 22 p^{5} T^{13} + 19 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 50 T^{2} - 48 T^{3} + 1314 T^{4} + 1656 T^{5} - 22624 T^{6} - 17136 T^{7} + 362831 T^{8} - 17136 p T^{9} - 22624 p^{2} T^{10} + 1656 p^{3} T^{11} + 1314 p^{4} T^{12} - 48 p^{5} T^{13} - 50 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 4 T - 38 T^{2} + 432 T^{3} + 178 T^{4} - 13412 T^{5} + 54432 T^{6} + 174676 T^{7} - 1768145 T^{8} + 174676 p T^{9} + 54432 p^{2} T^{10} - 13412 p^{3} T^{11} + 178 p^{4} T^{12} + 432 p^{5} T^{13} - 38 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 12 T + 10 T^{2} - 48 T^{3} + 78 p T^{4} + 2172 T^{5} - 115552 T^{6} - 337788 T^{7} + 529871 T^{8} - 337788 p T^{9} - 115552 p^{2} T^{10} + 2172 p^{3} T^{11} + 78 p^{5} T^{12} - 48 p^{5} T^{13} + 10 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 106 T^{2} - 80 T^{3} + 6122 T^{4} + 5720 T^{5} - 250256 T^{6} - 109200 T^{7} + 8837551 T^{8} - 109200 p T^{9} - 250256 p^{2} T^{10} + 5720 p^{3} T^{11} + 6122 p^{4} T^{12} - 80 p^{5} T^{13} - 106 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 12 T + 68 T^{2} - 52 T^{3} - 2014 T^{4} - 52 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 124 T^{2} + 840 T^{3} + 7718 T^{4} + 840 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 2 T - 143 T^{2} - 462 T^{3} + 11509 T^{4} + 35956 T^{5} - 616770 T^{6} - 834080 T^{7} + 28811410 T^{8} - 834080 p T^{9} - 616770 p^{2} T^{10} + 35956 p^{3} T^{11} + 11509 p^{4} T^{12} - 462 p^{5} T^{13} - 143 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 4 T - 126 T^{2} - 144 T^{3} + 10450 T^{4} + 29588 T^{5} - 461216 T^{6} - 16596 p T^{7} + 16456831 T^{8} - 16596 p^{2} T^{9} - 461216 p^{2} T^{10} + 29588 p^{3} T^{11} + 10450 p^{4} T^{12} - 144 p^{5} T^{13} - 126 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 4 T - 56 T^{2} + 184 T^{3} + 759 T^{4} + 184 p T^{5} - 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 20 T + 112 T^{2} - 616 T^{3} + 9154 T^{4} - 3628 T^{5} - 816160 T^{6} + 6913012 T^{7} - 44899389 T^{8} + 6913012 p T^{9} - 816160 p^{2} T^{10} - 3628 p^{3} T^{11} + 9154 p^{4} T^{12} - 616 p^{5} T^{13} + 112 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 8 T - 90 T^{2} + 1120 T^{3} + 314 T^{4} - 35992 T^{5} - 19824 T^{6} - 125560 T^{7} + 15058063 T^{8} - 125560 p T^{9} - 19824 p^{2} T^{10} - 35992 p^{3} T^{11} + 314 p^{4} T^{12} + 1120 p^{5} T^{13} - 90 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 4 T + 32 T^{2} - 12 p T^{3} + 5438 T^{4} - 12 p^{2} T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 16 T + 134 T^{2} - 2432 T^{3} + 25762 T^{4} - 178416 T^{5} + 2496832 T^{6} - 22590480 T^{7} + 136790271 T^{8} - 22590480 p T^{9} + 2496832 p^{2} T^{10} - 178416 p^{3} T^{11} + 25762 p^{4} T^{12} - 2432 p^{5} T^{13} + 134 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 22 T + 95 T^{2} - 894 T^{3} - 1575 T^{4} + 59112 T^{5} - 54650 T^{6} + 5002748 T^{7} + 123004082 T^{8} + 5002748 p T^{9} - 54650 p^{2} T^{10} + 59112 p^{3} T^{11} - 1575 p^{4} T^{12} - 894 p^{5} T^{13} + 95 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 36 T + 750 T^{2} + 10564 T^{3} + 110978 T^{4} + 10564 p T^{5} + 750 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 40 T + 686 T^{2} - 8960 T^{3} + 131138 T^{4} - 1660760 T^{5} + 16627776 T^{6} - 173801320 T^{7} + 1809325407 T^{8} - 173801320 p T^{9} + 16627776 p^{2} T^{10} - 1660760 p^{3} T^{11} + 131138 p^{4} T^{12} - 8960 p^{5} T^{13} + 686 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 26 T + 545 T^{2} + 7602 T^{3} + 85890 T^{4} + 7602 p T^{5} + 545 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.90197087621052174601491268932, −3.76389725070684186598297634500, −3.52063331187929457686622257379, −3.51415206954428380749684823314, −3.19919382746954116144772738026, −3.09254395603471887825800316230, −3.07428575773319411775007687217, −3.01531344685477473723423891232, −2.89502611656109036023381537583, −2.61446534234353864162439783016, −2.56183932602477452443813896045, −2.38456745802007765101027059369, −2.32430560168335318850680527149, −2.25120854891636335826180015051, −2.03350911094861567683376027530, −1.76561194909211296714125071193, −1.75193383268404646405830099782, −1.68704157344683643149480635699, −1.68303686551565594999127367243, −1.49875803878151169821230430118, −1.19459121441659270554653876078, −0.798743030076855565944921064766, −0.56365724433097074802322251908, −0.37826377385708166208727826888, −0.12960961069312011111755297147,
0.12960961069312011111755297147, 0.37826377385708166208727826888, 0.56365724433097074802322251908, 0.798743030076855565944921064766, 1.19459121441659270554653876078, 1.49875803878151169821230430118, 1.68303686551565594999127367243, 1.68704157344683643149480635699, 1.75193383268404646405830099782, 1.76561194909211296714125071193, 2.03350911094861567683376027530, 2.25120854891636335826180015051, 2.32430560168335318850680527149, 2.38456745802007765101027059369, 2.56183932602477452443813896045, 2.61446534234353864162439783016, 2.89502611656109036023381537583, 3.01531344685477473723423891232, 3.07428575773319411775007687217, 3.09254395603471887825800316230, 3.19919382746954116144772738026, 3.51415206954428380749684823314, 3.52063331187929457686622257379, 3.76389725070684186598297634500, 3.90197087621052174601491268932
Plot not available for L-functions of degree greater than 10.
