L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s − 5·11-s + 12-s + 13-s + 2·14-s + 15-s + 16-s − 7·17-s − 18-s + 20-s − 2·21-s + 5·22-s + 3·23-s − 24-s + 25-s − 26-s + 27-s − 2·28-s + 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s + 1.06·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.418888832\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.418888832\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.62075987766701, −12.11240283803357, −11.43594947245115, −10.85086633421344, −10.64966699575958, −10.23062905164899, −9.733306003383599, −9.265841896936243, −8.848860586699332, −8.490736151849587, −8.097240187873782, −7.308389573746952, −7.172035231586068, −6.560747145867678, −6.161198772324704, −5.510845926085967, −5.115570912080990, −4.386651115504842, −3.942183546979691, −3.151236193664942, −2.744225613693959, −2.263021166999354, −1.987117678273550, −0.8213454052628998, −0.5539783534487513,
0.5539783534487513, 0.8213454052628998, 1.987117678273550, 2.263021166999354, 2.744225613693959, 3.151236193664942, 3.942183546979691, 4.386651115504842, 5.115570912080990, 5.510845926085967, 6.161198772324704, 6.560747145867678, 7.172035231586068, 7.308389573746952, 8.097240187873782, 8.490736151849587, 8.848860586699332, 9.265841896936243, 9.733306003383599, 10.23062905164899, 10.64966699575958, 10.85086633421344, 11.43594947245115, 12.11240283803357, 12.62075987766701