L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s − 3·9-s + 2·10-s + 2·11-s + 16-s + 3·18-s + 2·19-s − 2·20-s − 2·22-s + 8·23-s − 25-s + 8·31-s − 32-s − 3·36-s + 4·37-s − 2·38-s + 2·40-s − 6·41-s + 4·43-s + 2·44-s + 6·45-s − 8·46-s − 8·47-s + 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 9-s + 0.632·10-s + 0.603·11-s + 1/4·16-s + 0.707·18-s + 0.458·19-s − 0.447·20-s − 0.426·22-s + 1.66·23-s − 1/5·25-s + 1.43·31-s − 0.176·32-s − 1/2·36-s + 0.657·37-s − 0.324·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s + 0.301·44-s + 0.894·45-s − 1.17·46-s − 1.16·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.082625972\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082625972\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.34230839159468, −14.75592594650029, −14.30850871623909, −13.67024738515228, −13.02874781792735, −12.34568030188076, −11.76580022483223, −11.44013750916585, −11.07440715741992, −10.36572008084008, −9.641412130520201, −9.222056430753970, −8.527301116840264, −8.213501140185126, −7.589971249714692, −6.986924459040909, −6.405974857468422, −5.813092783159750, −4.963501870855291, −4.439291957727134, −3.381670751029843, −3.156782315090820, −2.261968428663304, −1.219794263949905, −0.5036201812772420,
0.5036201812772420, 1.219794263949905, 2.261968428663304, 3.156782315090820, 3.381670751029843, 4.439291957727134, 4.963501870855291, 5.813092783159750, 6.405974857468422, 6.986924459040909, 7.589971249714692, 8.213501140185126, 8.527301116840264, 9.222056430753970, 9.641412130520201, 10.36572008084008, 11.07440715741992, 11.44013750916585, 11.76580022483223, 12.34568030188076, 13.02874781792735, 13.67024738515228, 14.30850871623909, 14.75592594650029, 15.34230839159468