L(s) = 1 | − 4·11-s − 13-s + 17-s + 4·19-s − 4·23-s − 6·29-s − 10·37-s + 2·41-s − 4·43-s + 8·47-s − 7·49-s − 10·53-s − 12·59-s − 2·61-s − 12·67-s + 2·73-s − 4·79-s + 12·83-s + 14·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 1.11·29-s − 1.64·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s − 49-s − 1.37·53-s − 1.56·59-s − 0.256·61-s − 1.46·67-s + 0.234·73-s − 0.450·79-s + 1.31·83-s + 1.48·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.80533208397275, −12.46908021911778, −11.91021294688932, −11.72025624693397, −10.94429371754954, −10.58428925118010, −10.35795105690192, −9.607207319901725, −9.413756415346795, −8.879077303271058, −8.204927500620574, −7.833382908321771, −7.506892809560981, −7.080540480001742, −6.360100798824519, −5.941155851920536, −5.415548240516145, −5.008535189622271, −4.609516106227255, −3.850758904955654, −3.335447842657815, −2.963296713203246, −2.256519296463093, −1.755467410546120, −1.151865969541291, 0, 0,
1.151865969541291, 1.755467410546120, 2.256519296463093, 2.963296713203246, 3.335447842657815, 3.850758904955654, 4.609516106227255, 5.008535189622271, 5.415548240516145, 5.941155851920536, 6.360100798824519, 7.080540480001742, 7.506892809560981, 7.833382908321771, 8.204927500620574, 8.879077303271058, 9.413756415346795, 9.607207319901725, 10.35795105690192, 10.58428925118010, 10.94429371754954, 11.72025624693397, 11.91021294688932, 12.46908021911778, 12.80533208397275