| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 4·10-s + 2·11-s + 13-s − 4·16-s + 19-s + 4·20-s − 4·22-s − 25-s − 2·26-s − 4·29-s + 9·31-s + 8·32-s + 3·37-s − 2·38-s + 10·41-s + 5·43-s + 4·44-s + 6·47-s + 2·50-s + 2·52-s − 12·53-s + 4·55-s + 8·58-s + 12·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s + 0.603·11-s + 0.277·13-s − 16-s + 0.229·19-s + 0.894·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.742·29-s + 1.61·31-s + 1.41·32-s + 0.493·37-s − 0.324·38-s + 1.56·41-s + 0.762·43-s + 0.603·44-s + 0.875·47-s + 0.282·50-s + 0.277·52-s − 1.64·53-s + 0.539·55-s + 1.05·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8379801548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8379801548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.87139152298459419633900996880, −9.879400023159387748546449740681, −9.459031124333216147077448358277, −8.583706845901838358159961836152, −7.67312246287158311998952783143, −6.65130832285966559117996014878, −5.72037257717675019727399870073, −4.23783192793666623809357478808, −2.43327032489084752763662196914, −1.15383453307155328641933378041,
1.15383453307155328641933378041, 2.43327032489084752763662196914, 4.23783192793666623809357478808, 5.72037257717675019727399870073, 6.65130832285966559117996014878, 7.67312246287158311998952783143, 8.583706845901838358159961836152, 9.459031124333216147077448358277, 9.879400023159387748546449740681, 10.87139152298459419633900996880
