| L(s) = 1 | − 2·2-s + 14·4-s − 10·5-s − 44·8-s + 20·10-s + 22·11-s + 64·13-s + 120·16-s + 64·17-s − 20·19-s − 140·20-s − 44·22-s + 84·23-s + 25·25-s − 128·26-s + 170·29-s + 258·31-s − 392·32-s − 128·34-s + 152·37-s + 40·38-s + 440·40-s + 578·41-s − 380·43-s + 308·44-s − 168·46-s + 484·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 7/4·4-s − 0.894·5-s − 1.94·8-s + 0.632·10-s + 0.603·11-s + 1.36·13-s + 15/8·16-s + 0.913·17-s − 0.241·19-s − 1.56·20-s − 0.426·22-s + 0.761·23-s + 1/5·25-s − 0.965·26-s + 1.08·29-s + 1.49·31-s − 2.16·32-s − 0.645·34-s + 0.675·37-s + 0.170·38-s + 1.73·40-s + 2.20·41-s − 1.34·43-s + 1.05·44-s − 0.538·46-s + 1.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.032528548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.032528548\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T - 5 p T^{2} - p^{2} T^{3} + 25 p^{2} T^{4} - p^{5} T^{5} - 5 p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 638 T^{2} + 289395 T^{4} - 638 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 p T + 53 T^{2} + 4462 p T^{3} - 2230004 T^{4} + 4462 p^{4} T^{5} + 53 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 64 T - 98 p T^{2} - 62464 T^{3} + 15011179 T^{4} - 62464 p^{3} T^{5} - 98 p^{7} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 32 T + 674 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T - 129 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 84 T - 15970 T^{2} + 109872 T^{3} + 296310435 T^{4} + 109872 p^{3} T^{5} - 15970 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 170 T - 875 p T^{2} - 934490 T^{3} + 1646110204 T^{4} - 934490 p^{3} T^{5} - 875 p^{7} T^{6} - 170 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 258 T - 9227 T^{2} - 4181922 T^{3} + 2873763876 T^{4} - 4181922 p^{3} T^{5} - 9227 p^{6} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 76 T + 37038 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 578 T + 119633 T^{2} - 44280002 T^{3} + 18213723796 T^{4} - 44280002 p^{3} T^{5} + 119633 p^{6} T^{6} - 578 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 380 T + 22294 T^{2} - 14025040 T^{3} - 2892298613 T^{4} - 14025040 p^{3} T^{5} + 22294 p^{6} T^{6} + 380 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 484 T - 31186 T^{2} - 27973264 T^{3} + 35359079347 T^{4} - 27973264 p^{3} T^{5} - 31186 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 544 T + 341738 T^{2} + 544 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 706 T - 34579 T^{2} - 86313442 T^{3} + 139556169340 T^{4} - 86313442 p^{3} T^{5} - 34579 p^{6} T^{6} - 706 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 668 T - 8702 T^{2} + 643952 T^{3} + 54152921371 T^{4} + 643952 p^{3} T^{5} - 8702 p^{6} T^{6} + 668 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1452 T + 987814 T^{2} - 753535728 T^{3} + 530939621979 T^{4} - 753535728 p^{3} T^{5} + 987814 p^{6} T^{6} - 1452 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 974 T + 837743 T^{2} + 974 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 1184 T + 1103106 T^{2} - 1184 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 408 T - 828782 T^{2} + 3740544 T^{3} + 665353900899 T^{4} + 3740544 p^{3} T^{5} - 828782 p^{6} T^{6} + 408 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 444 T + 84278 T^{2} + 457637904 T^{3} - 426923247237 T^{4} + 457637904 p^{3} T^{5} + 84278 p^{6} T^{6} - 444 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 513 T + p^{3} T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 668 T - 909878 T^{2} + 313454992 T^{3} + 598784032723 T^{4} + 313454992 p^{3} T^{5} - 909878 p^{6} T^{6} - 668 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.74202189957773208733922161351, −7.40675335572257749683085743070, −7.35795913310593485222133070998, −6.85876097421614074254475024440, −6.76837503013425195667533421007, −6.53671508768050018518937398296, −6.25668224776226682621264516553, −6.01184122223669254388450159259, −5.78834962370098925410542698939, −5.74234372994417939970504220329, −5.07246413061253096775066640046, −4.89955865502255272083603786556, −4.46631950805071963790934046175, −4.12383091107829019293933856820, −4.04990952262589589845853524545, −3.45752239275726090823309914421, −3.28978000979612565290508229440, −2.95461751725381566689677881459, −2.83735845998290954577301072006, −2.21792792774659061257923252139, −2.06883246623794529901977339246, −1.53167476644488562601502681461, −0.928527251024423362074489436654, −0.77321721684416384753616876487, −0.59654086940795914222944995266,
0.59654086940795914222944995266, 0.77321721684416384753616876487, 0.928527251024423362074489436654, 1.53167476644488562601502681461, 2.06883246623794529901977339246, 2.21792792774659061257923252139, 2.83735845998290954577301072006, 2.95461751725381566689677881459, 3.28978000979612565290508229440, 3.45752239275726090823309914421, 4.04990952262589589845853524545, 4.12383091107829019293933856820, 4.46631950805071963790934046175, 4.89955865502255272083603786556, 5.07246413061253096775066640046, 5.74234372994417939970504220329, 5.78834962370098925410542698939, 6.01184122223669254388450159259, 6.25668224776226682621264516553, 6.53671508768050018518937398296, 6.76837503013425195667533421007, 6.85876097421614074254475024440, 7.35795913310593485222133070998, 7.40675335572257749683085743070, 7.74202189957773208733922161351
