| L(s) = 1 | − 32·2-s + 162·3-s + 768·4-s − 1.25e3·5-s − 5.18e3·6-s + 4.80e3·7-s − 1.63e4·8-s + 1.96e4·9-s + 4.00e4·10-s + 3.73e4·11-s + 1.24e5·12-s − 8.61e4·13-s − 1.53e5·14-s − 2.02e5·15-s + 3.27e5·16-s − 3.07e5·17-s − 6.29e5·18-s − 1.80e5·19-s − 9.60e5·20-s + 7.77e5·21-s − 1.19e6·22-s + 3.71e5·23-s − 2.65e6·24-s + 1.17e6·25-s + 2.75e6·26-s + 2.12e6·27-s + 3.68e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s + 0.768·11-s + 1.73·12-s − 0.836·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.892·17-s − 1.41·18-s − 0.318·19-s − 1.34·20-s + 0.872·21-s − 1.08·22-s + 0.276·23-s − 1.63·24-s + 3/5·25-s + 1.18·26-s + 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 37322 T + 4848858478 T^{2} - 37322 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 86142 T + 17682364162 T^{2} + 86142 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 307304 T + 237625894798 T^{2} + 307304 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 180878 T + 471349366854 T^{2} + 180878 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 371024 T + 1598600740270 T^{2} - 371024 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4580976 T + 34233128110582 T^{2} - 4580976 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6783330 T + 58994392605742 T^{2} + 6783330 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 689460 T + 39939645043054 T^{2} - 689460 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1774112 T + 254796222952558 T^{2} + 1774112 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 52075012 T + 1682888090656022 T^{2} + 52075012 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2263820 T - 1217683269356066 T^{2} - 2263820 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 85320642 T + 4465696901992882 T^{2} + 85320642 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 46290908 T + 3269974886026294 T^{2} - 46290908 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11519952 T + 9626412682499158 T^{2} - 11519952 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 87057812 T + 4936001209469030 T^{2} - 87057812 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 427291306 T + 133008211351260646 T^{2} - 427291306 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 252566970 T + 129142718654320426 T^{2} + 252566970 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 669167320 T + 275112768869489438 T^{2} + 669167320 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 880993608 T + 564195902817295222 T^{2} - 880993608 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12654508 T - 51652739874329066 T^{2} - 12654508 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 298283058 T + 1350167803096458850 T^{2} - 298283058 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26280995261152712987303577124, −9.945057819210584668989814927445, −9.313281169840505457389635067561, −8.874958606631936111790943980814, −8.554669929690775621393978282971, −8.166095801804987508711674101776, −7.55187785051253990242050944398, −7.36939866648922497558090588887, −6.53111865106173224530097593357, −6.47503355987388045740357831308, −4.92510227385637604509560131619, −4.91519667045338823606098553148, −3.69184204390344104622993071687, −3.68977361999461565617641245199, −2.45634584007654463669253380225, −2.45511802813648306457228800907, −1.32874498404438742553697047419, −1.32742286471942929927520229593, 0, 0,
1.32742286471942929927520229593, 1.32874498404438742553697047419, 2.45511802813648306457228800907, 2.45634584007654463669253380225, 3.68977361999461565617641245199, 3.69184204390344104622993071687, 4.91519667045338823606098553148, 4.92510227385637604509560131619, 6.47503355987388045740357831308, 6.53111865106173224530097593357, 7.36939866648922497558090588887, 7.55187785051253990242050944398, 8.166095801804987508711674101776, 8.554669929690775621393978282971, 8.874958606631936111790943980814, 9.313281169840505457389635067561, 9.945057819210584668989814927445, 10.26280995261152712987303577124
