| L(s) = 1 | − 2·2-s + 3.20·3-s + 4·4-s + 7.27·5-s − 6.40·6-s + 7.24·7-s − 8·8-s − 16.7·9-s − 14.5·10-s + 10.2·11-s + 12.8·12-s − 64.0·13-s − 14.4·14-s + 23.2·15-s + 16·16-s + 20.5·17-s + 33.5·18-s + 41.6·19-s + 29.1·20-s + 23.1·21-s − 20.5·22-s + 23·23-s − 25.6·24-s − 72.0·25-s + 128.·26-s − 140.·27-s + 28.9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.615·3-s + 0.5·4-s + 0.650·5-s − 0.435·6-s + 0.391·7-s − 0.353·8-s − 0.620·9-s − 0.460·10-s + 0.281·11-s + 0.307·12-s − 1.36·13-s − 0.276·14-s + 0.400·15-s + 0.250·16-s + 0.293·17-s + 0.438·18-s + 0.503·19-s + 0.325·20-s + 0.240·21-s − 0.198·22-s + 0.208·23-s − 0.217·24-s − 0.576·25-s + 0.966·26-s − 0.998·27-s + 0.195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 3.20T + 27T^{2} \) |
| 5 | \( 1 - 7.27T + 125T^{2} \) |
| 7 | \( 1 - 7.24T + 343T^{2} \) |
| 11 | \( 1 - 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.6T + 6.85e3T^{2} \) |
| 31 | \( 1 - 48.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 199.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 58.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 157.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 642.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 643.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 790.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 996.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 540.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 122.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 911.T + 9.12e5T^{2} \) |
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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
−9.076805484138674217658786668425, −7.911786733571271237989538329304, −7.63474759271664687551673424895, −6.43974376459927221682564269010, −5.61532033409135666445827106637, −4.64404308768361171275937099887, −3.21532715117137195786587698872, −2.42376966336641794922593498558, −1.48747347237023751010950706638, 0,
1.48747347237023751010950706638, 2.42376966336641794922593498558, 3.21532715117137195786587698872, 4.64404308768361171275937099887, 5.61532033409135666445827106637, 6.43974376459927221682564269010, 7.63474759271664687551673424895, 7.911786733571271237989538329304, 9.076805484138674217658786668425
