| L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s + 6·11-s + 2·13-s + 2·14-s + 16-s − 2·19-s + 20-s − 6·22-s − 6·23-s + 25-s − 2·26-s − 2·28-s + 6·29-s + 8·31-s − 32-s − 2·35-s − 2·37-s + 2·38-s − 40-s + 6·41-s + 6·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.458·19-s + 0.223·20-s − 1.27·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.338·35-s − 0.328·37-s + 0.324·38-s − 0.158·40-s + 0.937·41-s + 0.904·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801132432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801132432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.34470789165415, −12.53769023946333, −12.23798724954810, −11.87138774046706, −11.32139557338935, −10.76766517760771, −10.27152755954208, −9.826402980530766, −9.411247985820962, −9.061539470314906, −8.433160103014004, −8.144225852626397, −7.445278978842722, −6.623169095825274, −6.480055179504648, −6.225832893837065, −5.570304332708968, −4.715900251111513, −4.096125309155228, −3.739740393266973, −2.941969099557816, −2.509792739395913, −1.571182272767617, −1.298741423817473, −0.4550839174428830,
0.4550839174428830, 1.298741423817473, 1.571182272767617, 2.509792739395913, 2.941969099557816, 3.739740393266973, 4.096125309155228, 4.715900251111513, 5.570304332708968, 6.225832893837065, 6.480055179504648, 6.623169095825274, 7.445278978842722, 8.144225852626397, 8.433160103014004, 9.061539470314906, 9.411247985820962, 9.826402980530766, 10.27152755954208, 10.76766517760771, 11.32139557338935, 11.87138774046706, 12.23798724954810, 12.53769023946333, 13.34470789165415
