| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 3·11-s − 13-s + 4·14-s + 16-s + 19-s + 20-s + 3·22-s − 7·23-s + 25-s + 26-s − 4·28-s + 29-s − 3·31-s − 32-s − 4·35-s − 11·37-s − 38-s − 40-s + 4·41-s + 7·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.229·19-s + 0.223·20-s + 0.639·22-s − 1.45·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s − 0.676·35-s − 1.80·37-s − 0.162·38-s − 0.158·40-s + 0.624·41-s + 1.06·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7408044410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7408044410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.62995098749685, −12.19347367380429, −11.66633315720988, −11.13622566145837, −10.39428603407375, −10.27787968318554, −9.984728998166158, −9.357698392545713, −9.061747628363141, −8.547376719225322, −7.997524757603886, −7.445761991714181, −7.089751346205784, −6.618150532504295, −6.020799156934736, −5.693345899990142, −5.294613892975990, −4.448892750294649, −3.901789145453918, −3.213651543925819, −2.950681989434999, −2.114469183730555, −1.979270671177947, −0.8685563246469453, −0.3028723586519543,
0.3028723586519543, 0.8685563246469453, 1.979270671177947, 2.114469183730555, 2.950681989434999, 3.213651543925819, 3.901789145453918, 4.448892750294649, 5.294613892975990, 5.693345899990142, 6.020799156934736, 6.618150532504295, 7.089751346205784, 7.445761991714181, 7.997524757603886, 8.547376719225322, 9.061747628363141, 9.357698392545713, 9.984728998166158, 10.27787968318554, 10.39428603407375, 11.13622566145837, 11.66633315720988, 12.19347367380429, 12.62995098749685
