| L(s) = 1 | + 0.0822·2-s + 1.81·3-s − 1.99·4-s + 4.31·5-s + 0.149·6-s + 0.690·7-s − 0.328·8-s + 0.306·9-s + 0.355·10-s − 2.95·11-s − 3.62·12-s − 1.93·13-s + 0.0567·14-s + 7.85·15-s + 3.95·16-s − 2.07·17-s + 0.0251·18-s + 3.74·19-s − 8.61·20-s + 1.25·21-s − 0.243·22-s + 4.34·23-s − 0.597·24-s + 13.6·25-s − 0.159·26-s − 4.89·27-s − 1.37·28-s + ⋯ |
| L(s) = 1 | + 0.0581·2-s + 1.04·3-s − 0.996·4-s + 1.93·5-s + 0.0610·6-s + 0.261·7-s − 0.116·8-s + 0.102·9-s + 0.112·10-s − 0.891·11-s − 1.04·12-s − 0.536·13-s + 0.0151·14-s + 2.02·15-s + 0.989·16-s − 0.503·17-s + 0.00593·18-s + 0.859·19-s − 1.92·20-s + 0.273·21-s − 0.0518·22-s + 0.906·23-s − 0.121·24-s + 2.73·25-s − 0.0312·26-s − 0.942·27-s − 0.260·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.768044603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768044603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0822T + 2T^{2} \) |
| 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 - 0.690T + 7T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 9.72T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 - 6.71T + 47T^{2} \) |
| 53 | \( 1 + 0.0484T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 + 0.964T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−12.66564910999546413338230367285, −10.84372354517246757988222011113, −9.814551353025877531911532599420, −9.260607331125647366210656839946, −8.554619911522751292012217507709, −7.29730184639615918787944248904, −5.64315272705332536570821875323, −5.03459987685295897876657547296, −3.19212418361870217323656071858, −1.99248029613211426635712683195,
1.99248029613211426635712683195, 3.19212418361870217323656071858, 5.03459987685295897876657547296, 5.64315272705332536570821875323, 7.29730184639615918787944248904, 8.554619911522751292012217507709, 9.260607331125647366210656839946, 9.814551353025877531911532599420, 10.84372354517246757988222011113, 12.66564910999546413338230367285
