| L(s) = 1 | − 2·4-s + 5-s + 2·7-s − 11-s + 4·16-s + 17-s + 19-s − 2·20-s − 8·23-s + 25-s − 4·28-s − 5·29-s + 2·35-s + 6·37-s + 9·41-s − 11·43-s + 2·44-s − 6·47-s − 3·49-s − 4·53-s − 55-s − 10·59-s − 8·64-s − 3·67-s − 2·68-s − 12·73-s − 2·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.755·7-s − 0.301·11-s + 16-s + 0.242·17-s + 0.229·19-s − 0.447·20-s − 1.66·23-s + 1/5·25-s − 0.755·28-s − 0.928·29-s + 0.338·35-s + 0.986·37-s + 1.40·41-s − 1.67·43-s + 0.301·44-s − 0.875·47-s − 3/7·49-s − 0.549·53-s − 0.134·55-s − 1.30·59-s − 64-s − 0.366·67-s − 0.242·68-s − 1.40·73-s − 0.229·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034661198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034661198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.39799279330668, −13.19514607054432, −12.60807267520730, −12.05107607564288, −11.66912410433472, −10.95698524135248, −10.63773059427990, −9.902221404817720, −9.623129094559667, −9.267251973856084, −8.493295321636669, −8.090967712932062, −7.804106626465710, −7.213022940316016, −6.312125931421792, −5.939694095160534, −5.413842252650062, −4.890706107423827, −4.410681602253026, −3.896227716590950, −3.230236096392749, −2.568644149619767, −1.702364886759711, −1.373162924840656, −0.3114570159035682,
0.3114570159035682, 1.373162924840656, 1.702364886759711, 2.568644149619767, 3.230236096392749, 3.896227716590950, 4.410681602253026, 4.890706107423827, 5.413842252650062, 5.939694095160534, 6.312125931421792, 7.213022940316016, 7.804106626465710, 8.090967712932062, 8.493295321636669, 9.267251973856084, 9.623129094559667, 9.902221404817720, 10.63773059427990, 10.95698524135248, 11.66912410433472, 12.05107607564288, 12.60807267520730, 13.19514607054432, 13.39799279330668
