| L(s) = 1 | − 2.09·2-s + 1.02·3-s + 2.36·4-s + 3.67·5-s − 2.14·6-s + 2.35·7-s − 0.772·8-s − 1.94·9-s − 7.68·10-s + 11-s + 2.42·12-s + 0.397·13-s − 4.92·14-s + 3.76·15-s − 3.12·16-s − 1.34·17-s + 4.07·18-s + 4.83·19-s + 8.70·20-s + 2.41·21-s − 2.09·22-s − 2.43·23-s − 0.791·24-s + 8.51·25-s − 0.829·26-s − 5.07·27-s + 5.58·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 0.591·3-s + 1.18·4-s + 1.64·5-s − 0.874·6-s + 0.891·7-s − 0.273·8-s − 0.649·9-s − 2.42·10-s + 0.301·11-s + 0.701·12-s + 0.110·13-s − 1.31·14-s + 0.972·15-s − 0.781·16-s − 0.327·17-s + 0.960·18-s + 1.11·19-s + 1.94·20-s + 0.527·21-s − 0.445·22-s − 0.507·23-s − 0.161·24-s + 1.70·25-s − 0.162·26-s − 0.976·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.480632756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480632756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 - 1.02T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 13 | \( 1 - 0.397T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 - 4.97T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.66T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 - 2.00T + 73T^{2} \) |
| 79 | \( 1 - 7.47T + 79T^{2} \) |
| 83 | \( 1 + 2.98T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.240754732864233683314083759133, −9.036963028730054877090993502074, −8.118419130885524194596787936364, −7.49642376080899259823826568448, −6.37712591620939234881916617015, −5.63826846487616276701554679045, −4.56219738448514130683969981323, −2.85073049964112147158837405463, −2.03476333045085613252670757819, −1.16448818337645602186047679856,
1.16448818337645602186047679856, 2.03476333045085613252670757819, 2.85073049964112147158837405463, 4.56219738448514130683969981323, 5.63826846487616276701554679045, 6.37712591620939234881916617015, 7.49642376080899259823826568448, 8.118419130885524194596787936364, 9.036963028730054877090993502074, 9.240754732864233683314083759133
