| L(s) = 1 | + 4.17·2-s + 5.44·3-s + 9.42·4-s + 5·5-s + 22.7·6-s − 28.9·7-s + 5.96·8-s + 2.63·9-s + 20.8·10-s − 11·11-s + 51.3·12-s − 30.1·13-s − 120.·14-s + 27.2·15-s − 50.5·16-s − 74.3·17-s + 11.0·18-s − 19·19-s + 47.1·20-s − 157.·21-s − 45.9·22-s − 168.·23-s + 32.4·24-s + 25·25-s − 125.·26-s − 132.·27-s − 273.·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.04·3-s + 1.17·4-s + 0.447·5-s + 1.54·6-s − 1.56·7-s + 0.263·8-s + 0.0976·9-s + 0.660·10-s − 0.301·11-s + 1.23·12-s − 0.642·13-s − 2.30·14-s + 0.468·15-s − 0.789·16-s − 1.06·17-s + 0.144·18-s − 0.229·19-s + 0.527·20-s − 1.63·21-s − 0.445·22-s − 1.53·23-s + 0.276·24-s + 0.200·25-s − 0.948·26-s − 0.945·27-s − 1.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 4.17T + 8T^{2} \) |
| 3 | \( 1 - 5.44T + 27T^{2} \) |
| 7 | \( 1 + 28.9T + 343T^{2} \) |
| 13 | \( 1 + 30.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 168.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 148.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 853.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 290.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 902.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 495.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 377.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 812.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 674.T + 9.12e5T^{2} \) |
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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
−9.337405651417663876969363776705, −8.293359242552491624335042049312, −7.19621347073305147881785207826, −6.25387590134033515609211641741, −5.80681353881083075711189578604, −4.46486220696661772115348709228, −3.75922834277785029325164035039, −2.63648889862461958844337885012, −2.46134250376824087078436661165, 0,
2.46134250376824087078436661165, 2.63648889862461958844337885012, 3.75922834277785029325164035039, 4.46486220696661772115348709228, 5.80681353881083075711189578604, 6.25387590134033515609211641741, 7.19621347073305147881785207826, 8.293359242552491624335042049312, 9.337405651417663876969363776705
