| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 6·9-s − 2·10-s − 13-s − 16-s + 3·17-s + 6·18-s − 2·20-s + 3·25-s + 26-s − 4·29-s − 5·32-s − 3·34-s + 6·36-s − 4·37-s + 6·40-s − 12·41-s − 12·45-s + 2·49-s − 3·50-s + 52-s − 20·53-s + 4·58-s + 12·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 2·9-s − 0.632·10-s − 0.277·13-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 0.447·20-s + 3/5·25-s + 0.196·26-s − 0.742·29-s − 0.883·32-s − 0.514·34-s + 36-s − 0.657·37-s + 0.948·40-s − 1.87·41-s − 1.78·45-s + 2/7·49-s − 0.424·50-s + 0.138·52-s − 2.74·53-s + 0.525·58-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.445370946078724095421771046114, −8.908301443936015740885982494409, −8.540017954936382733473658077211, −8.057484724505346957367765946867, −7.68505998628538599450969837444, −6.83549653004652567501187752741, −6.35657857212787094671023234155, −5.67610609191103462331322799059, −5.20829761143314977818908810247, −4.96837461124932327787635715000, −3.80580534244267015608673762747, −3.21221371525954144695929729012, −2.42012886642585914453369006792, −1.49982897493095869103804944828, 0,
1.49982897493095869103804944828, 2.42012886642585914453369006792, 3.21221371525954144695929729012, 3.80580534244267015608673762747, 4.96837461124932327787635715000, 5.20829761143314977818908810247, 5.67610609191103462331322799059, 6.35657857212787094671023234155, 6.83549653004652567501187752741, 7.68505998628538599450969837444, 8.057484724505346957367765946867, 8.540017954936382733473658077211, 8.908301443936015740885982494409, 9.445370946078724095421771046114
