| L(s) = 1 | − 2·3-s − 4-s − 3·5-s + 3·7-s + 9-s + 2·12-s + 2·13-s + 6·15-s − 3·16-s − 17-s + 6·19-s + 3·20-s − 6·21-s − 6·23-s + 2·25-s + 4·27-s − 3·28-s − 29-s − 14·31-s − 9·35-s − 36-s − 8·37-s − 4·39-s − 12·41-s − 4·43-s − 3·45-s + 7·47-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 1.34·5-s + 1.13·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 1.54·15-s − 3/4·16-s − 0.242·17-s + 1.37·19-s + 0.670·20-s − 1.30·21-s − 1.25·23-s + 2/5·25-s + 0.769·27-s − 0.566·28-s − 0.185·29-s − 2.51·31-s − 1.52·35-s − 1/6·36-s − 1.31·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s − 0.447·45-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20205 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20205 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 449 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 24 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 96 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 98 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 104 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 60 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−16.0101883773, −15.7127952957, −15.1307093590, −14.4833504461, −14.1416115132, −13.6444032739, −13.0745179905, −12.3417619265, −11.9675072572, −11.6015282530, −11.1701127784, −10.9583286488, −10.2088799181, −9.56698140985, −8.72140922384, −8.49937604267, −7.79940980152, −7.27238448463, −6.75631526528, −5.78375798319, −5.25530446472, −4.81619744579, −3.97716358034, −3.49496342285, −1.75368043771, 0,
1.75368043771, 3.49496342285, 3.97716358034, 4.81619744579, 5.25530446472, 5.78375798319, 6.75631526528, 7.27238448463, 7.79940980152, 8.49937604267, 8.72140922384, 9.56698140985, 10.2088799181, 10.9583286488, 11.1701127784, 11.6015282530, 11.9675072572, 12.3417619265, 13.0745179905, 13.6444032739, 14.1416115132, 14.4833504461, 15.1307093590, 15.7127952957, 16.0101883773
