| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 4·10-s − 12·13-s − 4·16-s − 6·17-s − 4·20-s − 7·25-s − 24·26-s + 20·29-s − 8·32-s − 12·34-s + 16·37-s + 16·41-s − 5·49-s − 14·50-s − 24·52-s + 12·53-s + 40·58-s + 14·61-s − 8·64-s + 24·65-s − 12·68-s − 22·73-s + 32·74-s + 8·80-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s − 3.32·13-s − 16-s − 1.45·17-s − 0.894·20-s − 7/5·25-s − 4.70·26-s + 3.71·29-s − 1.41·32-s − 2.05·34-s + 2.63·37-s + 2.49·41-s − 5/7·49-s − 1.97·50-s − 3.32·52-s + 1.64·53-s + 5.25·58-s + 1.79·61-s − 64-s + 2.97·65-s − 1.45·68-s − 2.57·73-s + 3.71·74-s + 0.894·80-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
| \(L(1)\) |
\(\approx\) |
\(1.811772200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.811772200\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.513296718878561122779255009178, −7.935555959343104225865564799258, −7.33179869161333338136909211767, −7.32016227740693371415400237232, −6.66260307528373962910372616260, −6.12831923497636436478284774657, −5.74803450606645387288578022188, −4.85468716698404797055779902636, −4.75227634922127063518685270995, −4.20883425493377341015603476651, −4.12511627512637579479297949689, −2.90182171573806321583393234308, −2.47342224903748840497823197638, −2.44031011733442394942262196334, −0.53708427580882030700657091576,
0.53708427580882030700657091576, 2.44031011733442394942262196334, 2.47342224903748840497823197638, 2.90182171573806321583393234308, 4.12511627512637579479297949689, 4.20883425493377341015603476651, 4.75227634922127063518685270995, 4.85468716698404797055779902636, 5.74803450606645387288578022188, 6.12831923497636436478284774657, 6.66260307528373962910372616260, 7.32016227740693371415400237232, 7.33179869161333338136909211767, 7.935555959343104225865564799258, 8.513296718878561122779255009178
