| L(s) = 1 | + 20·7-s + 440·13-s − 217·16-s − 4.76e3·31-s − 7.24e3·37-s + 5.24e3·43-s + 200·49-s − 5.51e3·61-s − 7.48e3·67-s + 1.56e4·73-s + 8.80e3·91-s + 2.88e4·97-s − 1.23e4·103-s − 4.34e3·112-s + 1.93e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.68e4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.408·7-s + 2.60·13-s − 0.847·16-s − 4.96·31-s − 5.28·37-s + 2.83·43-s + 0.0832·49-s − 1.48·61-s − 1.66·67-s + 2.93·73-s + 1.06·91-s + 3.06·97-s − 1.16·103-s − 0.345·112-s + 0.132·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.38·169-s + 3.34e−5·173-s + 3.12e−5·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.345709159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.345709159\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 217 T^{4} + p^{16} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 p^{2} T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 220 T + 24200 T^{2} - 220 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 13944833602 T^{4} + p^{16} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 221438 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 156488845118 T^{4} + p^{16} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 422312 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1192 T + p^{4} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 3620 T + 6552200 T^{2} + 3620 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3250522 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2620 T + 3432200 T^{2} - 2620 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 47566111009598 T^{4} + p^{16} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 13468208583998 T^{4} + p^{16} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 23242472 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1378 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 3740 T + 6993800 T^{2} + 3740 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 3299362 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7810 T + 30498050 T^{2} - 7810 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 77525618 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 2054763907459838 T^{4} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 4115518 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 14410 T + 103824050 T^{2} - 14410 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.082964905467435183558852055356, −8.018576449231102748228995058995, −7.931517789659263607822022124302, −7.15094124847064514632508606098, −7.12107350067896561015858819567, −6.96368610485236476039723476628, −6.94446578832028809204334878653, −6.09628787008882391972665008484, −6.07271659514786279774646776461, −5.76951227110850092699285286471, −5.47862155618125170467550052492, −5.27220357531126172193797039576, −4.99081910937806796685023704253, −4.33350188752537360345763751689, −4.29191934406826751760201761359, −3.80734811609662071457316636233, −3.44361084665461022945015748755, −3.33884682211530523870258154607, −3.19424085753658841147759565359, −2.09470845550956436472553503583, −1.91488105516431786396538337777, −1.82287664183579587638260922976, −1.38625729665413182759686939553, −0.61594116163256906557998023514, −0.35404012861708018225072162093,
0.35404012861708018225072162093, 0.61594116163256906557998023514, 1.38625729665413182759686939553, 1.82287664183579587638260922976, 1.91488105516431786396538337777, 2.09470845550956436472553503583, 3.19424085753658841147759565359, 3.33884682211530523870258154607, 3.44361084665461022945015748755, 3.80734811609662071457316636233, 4.29191934406826751760201761359, 4.33350188752537360345763751689, 4.99081910937806796685023704253, 5.27220357531126172193797039576, 5.47862155618125170467550052492, 5.76951227110850092699285286471, 6.07271659514786279774646776461, 6.09628787008882391972665008484, 6.94446578832028809204334878653, 6.96368610485236476039723476628, 7.12107350067896561015858819567, 7.15094124847064514632508606098, 7.931517789659263607822022124302, 8.018576449231102748228995058995, 8.082964905467435183558852055356
