L(s) = 1 | − 2·7-s + 4·19-s + 4·25-s + 12·29-s + 16·31-s − 8·37-s + 24·47-s − 3·49-s − 12·53-s + 24·59-s − 32·103-s − 4·109-s − 12·113-s + 16·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 8·175-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.917·19-s + 4/5·25-s + 2.22·29-s + 2.87·31-s − 1.31·37-s + 3.50·47-s − 3/7·49-s − 1.64·53-s + 3.12·59-s − 3.15·103-s − 0.383·109-s − 1.12·113-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s − 0.604·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.132667910\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.132667910\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−8.503764164737708725993132932199, −8.469204947544295420021501899176, −7.900989008141513844519779175484, −7.53688459541587718216138216824, −7.04553103045493731933468230575, −6.79004559055561789030135910470, −6.37479369745579898372395713839, −6.28730855364845039368042757494, −5.63001794155721557041355695650, −5.10885848877651099318950978038, −5.08551732690939680305511450819, −4.40567675297270199358108154002, −4.05633313539108606858140977856, −3.68441074831981262996201358204, −2.95363662569522248918609961315, −2.71239194733128106647376662670, −2.59502862412262800116140544329, −1.59666040841639956781098124158, −0.947852183210974502938628026102, −0.65312076414784849846568338763,
0.65312076414784849846568338763, 0.947852183210974502938628026102, 1.59666040841639956781098124158, 2.59502862412262800116140544329, 2.71239194733128106647376662670, 2.95363662569522248918609961315, 3.68441074831981262996201358204, 4.05633313539108606858140977856, 4.40567675297270199358108154002, 5.08551732690939680305511450819, 5.10885848877651099318950978038, 5.63001794155721557041355695650, 6.28730855364845039368042757494, 6.37479369745579898372395713839, 6.79004559055561789030135910470, 7.04553103045493731933468230575, 7.53688459541587718216138216824, 7.900989008141513844519779175484, 8.469204947544295420021501899176, 8.503764164737708725993132932199