| L(s) = 1 | − 18·5-s − 2·7-s + 6·9-s − 18·11-s − 10·13-s + 40·19-s − 18·23-s + 139·25-s + 18·29-s − 38·31-s + 36·35-s + 128·37-s − 126·41-s + 46·43-s − 108·45-s − 54·47-s + 45·49-s + 324·55-s − 126·59-s + 62·61-s − 12·63-s + 180·65-s + 106·67-s − 208·73-s + 36·77-s − 14·79-s − 45·81-s + ⋯ |
| L(s) = 1 | − 3.59·5-s − 2/7·7-s + 2/3·9-s − 1.63·11-s − 0.769·13-s + 2.10·19-s − 0.782·23-s + 5.55·25-s + 0.620·29-s − 1.22·31-s + 1.02·35-s + 3.45·37-s − 3.07·41-s + 1.06·43-s − 2.39·45-s − 1.14·47-s + 0.918·49-s + 5.89·55-s − 2.13·59-s + 1.01·61-s − 0.190·63-s + 2.76·65-s + 1.58·67-s − 2.84·73-s + 0.467·77-s − 0.177·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2438540205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2438540205\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 p T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 2 T - 41 T^{2} - 106 T^{3} - 572 T^{4} - 106 p^{2} T^{5} - 41 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 18 T + 359 T^{2} + 4518 T^{3} + 61428 T^{4} + 4518 p^{2} T^{5} + 359 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 10 T - 47 T^{2} - 1910 T^{3} - 23852 T^{4} - 1910 p^{2} T^{5} - 47 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 20 T + 606 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18 T + 1175 T^{2} + 19206 T^{3} + 915780 T^{4} + 19206 p^{2} T^{5} + 1175 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 18 T + 1745 T^{2} - 29466 T^{3} + 2063316 T^{4} - 29466 p^{2} T^{5} + 1745 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 38 T - 353 T^{2} - 4750 T^{3} + 918004 T^{4} - 4750 p^{2} T^{5} - 353 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 508662 p^{2} T^{5} + 9329 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 46 T - 1625 T^{2} - 46 p T^{3} + 3604 p^{2} T^{4} - 46 p^{3} T^{5} - 1625 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 54 T + 4751 T^{2} + 204066 T^{3} + 11548308 T^{4} + 204066 p^{2} T^{5} + 4751 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 126 T + 10535 T^{2} + 660618 T^{3} + 33793140 T^{4} + 660618 p^{2} T^{5} + 10535 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 60946 p^{2} T^{5} - 2615 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 106 T - 65 T^{2} - 246238 T^{3} + 57123076 T^{4} - 246238 p^{2} T^{5} - 65 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 14 T - 10985 T^{2} - 18214 T^{3} + 84841444 T^{4} - 18214 p^{2} T^{5} - 10985 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 378 T + 72863 T^{2} - 9538830 T^{3} + 917456196 T^{4} - 9538830 p^{2} T^{5} + 72863 p^{4} T^{6} - 378 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 147490 p^{2} T^{5} - 8087 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−9.360792969584319585403220389594, −9.088942613404189964053084332492, −8.851959238502752369320578238840, −8.086147724185144064846346225606, −8.031322823126616782190067622453, −7.983251700254422522710477324063, −7.83479549471841972144866279190, −7.57719767346730460336213020282, −7.28474345165809796533211691652, −6.89168188208284409090052918462, −6.83878757895171972935704247588, −6.24914464636957940769422695232, −5.75716243159949674768872300727, −5.35216484770036810621889406606, −5.22030790239369887826294440978, −4.56958643147551926490471684925, −4.45162547544470042864514415365, −4.24640654739287124804145749730, −3.70185427456299874378854740897, −3.32261068894178789500529427228, −3.23886559272518401240426636071, −2.69396914812535854176709091210, −2.02762009226977346469245053078, −0.925777670434345044508533533931, −0.23865681734576980419040911121,
0.23865681734576980419040911121, 0.925777670434345044508533533931, 2.02762009226977346469245053078, 2.69396914812535854176709091210, 3.23886559272518401240426636071, 3.32261068894178789500529427228, 3.70185427456299874378854740897, 4.24640654739287124804145749730, 4.45162547544470042864514415365, 4.56958643147551926490471684925, 5.22030790239369887826294440978, 5.35216484770036810621889406606, 5.75716243159949674768872300727, 6.24914464636957940769422695232, 6.83878757895171972935704247588, 6.89168188208284409090052918462, 7.28474345165809796533211691652, 7.57719767346730460336213020282, 7.83479549471841972144866279190, 7.983251700254422522710477324063, 8.031322823126616782190067622453, 8.086147724185144064846346225606, 8.851959238502752369320578238840, 9.088942613404189964053084332492, 9.360792969584319585403220389594
