| L(s) = 1 | − 2-s − 4-s + 4·5-s + 3·8-s − 4·10-s + 12·13-s − 16-s + 12·17-s − 4·20-s + 2·25-s − 12·26-s − 4·29-s − 5·32-s − 12·34-s − 20·37-s + 12·40-s + 4·41-s − 14·49-s − 2·50-s − 12·52-s + 12·53-s + 4·58-s − 4·61-s + 7·64-s + 48·65-s − 12·68-s + 20·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s + 1.06·8-s − 1.26·10-s + 3.32·13-s − 1/4·16-s + 2.91·17-s − 0.894·20-s + 2/5·25-s − 2.35·26-s − 0.742·29-s − 0.883·32-s − 2.05·34-s − 3.28·37-s + 1.89·40-s + 0.624·41-s − 2·49-s − 0.282·50-s − 1.66·52-s + 1.64·53-s + 0.525·58-s − 0.512·61-s + 7/8·64-s + 5.95·65-s − 1.45·68-s + 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.176459332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.176459332\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.700339636508216566718463872719, −8.062008804214737453511932083588, −7.962026255736907116683171399764, −7.33667434445098454643296446245, −6.39584814801769229586264060019, −6.37820814688737049661034336254, −5.69178884171323828176864961556, −5.30285359031178015236684552035, −5.22523151161492305197654101068, −3.84859941092912106278451327058, −3.69904169102384607719527879903, −3.25148783575260438600086430315, −1.95255670636420425042320954209, −1.48488050469889056862198933364, −1.07628396925620245570845051253,
1.07628396925620245570845051253, 1.48488050469889056862198933364, 1.95255670636420425042320954209, 3.25148783575260438600086430315, 3.69904169102384607719527879903, 3.84859941092912106278451327058, 5.22523151161492305197654101068, 5.30285359031178015236684552035, 5.69178884171323828176864961556, 6.37820814688737049661034336254, 6.39584814801769229586264060019, 7.33667434445098454643296446245, 7.962026255736907116683171399764, 8.062008804214737453511932083588, 8.700339636508216566718463872719
