| 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 + 1.60e4·11-s − 1.24e5·12-s + 6.29e4·13-s − 1.53e5·14-s − 2.02e5·15-s + 3.27e5·16-s − 1.70e4·17-s − 6.29e5·18-s − 2.70e5·19-s + 9.60e5·20-s − 7.77e5·21-s − 5.15e5·22-s − 2.17e6·23-s + 2.65e6·24-s + 1.17e6·25-s − 2.01e6·26-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 + 0.331·11-s − 1.73·12-s + 0.611·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.0495·17-s − 1.41·18-s − 0.476·19-s + 1.34·20-s − 0.872·21-s − 0.468·22-s − 1.62·23-s + 1.63·24-s + 3/5·25-s − 0.864·26-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 | $D_{4}$ | \( 1 - 16098 T + 4015956814 T^{2} - 16098 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 62986 T + 14425332594 T^{2} - 62986 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 17076 T + 78342667702 T^{2} + 17076 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 270482 T + 241685167614 T^{2} + 270482 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2175324 T + 4684013073070 T^{2} + 2175324 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 146124 p T + 33290852553646 T^{2} + 146124 p^{10} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6996002 T + 63495769563582 T^{2} + 6996002 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7489760 T + 130757454598758 T^{2} + 7489760 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8148024 T + 526417754408350 T^{2} + 8148024 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8115160 T + 573428391020070 T^{2} - 8115160 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11429832 T + 1120561416416734 T^{2} - 11429832 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 100851414 T + 8539797839558506 T^{2} - 100851414 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 71517396 T + 18173652162936982 T^{2} + 71517396 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 174439444 T + 28291208886333582 T^{2} - 174439444 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 171733144 T + 61304988690275478 T^{2} - 171733144 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 212488326 T + 102802074682452910 T^{2} - 212488326 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 115112254 T + 27621411334280826 T^{2} - 115112254 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 305780576 T + 177122319109850526 T^{2} + 305780576 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 303202800 T + 353106954701736982 T^{2} + 303202800 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 637798260 T + 323359141171120918 T^{2} - 637798260 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 176146618 T + 1361067539765786034 T^{2} - 176146618 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−10.31988576058209795047908607942, −10.21881246787961768796208509195, −9.377399599418267945518782262325, −9.277709110018772392776504729668, −8.344502817719688506973769599831, −8.309394161089333346364202856925, −7.24006770262581350147847226340, −7.21424761040813545443579873979, −6.23889392355826808499171835572, −6.17984105586084881126336452959, −5.30368417778411084650069899967, −5.25415081739254887132645624079, −3.87523164881449737592643246149, −3.84982056176413042640517274902, −2.24015170457197760922465248312, −2.21899200437104060275368024618, −1.24249214828933420685683259011, −1.22727046306173138343320515273, 0, 0,
1.22727046306173138343320515273, 1.24249214828933420685683259011, 2.21899200437104060275368024618, 2.24015170457197760922465248312, 3.84982056176413042640517274902, 3.87523164881449737592643246149, 5.25415081739254887132645624079, 5.30368417778411084650069899967, 6.17984105586084881126336452959, 6.23889392355826808499171835572, 7.21424761040813545443579873979, 7.24006770262581350147847226340, 8.309394161089333346364202856925, 8.344502817719688506973769599831, 9.277709110018772392776504729668, 9.377399599418267945518782262325, 10.21881246787961768796208509195, 10.31988576058209795047908607942
