| L(s) = 1 | + 2.96·3-s − 0.378·5-s − 1.65·7-s + 5.76·9-s − 0.346·13-s − 1.12·15-s − 7.53·17-s − 4.76·19-s − 4.89·21-s − 7.88·23-s − 4.85·25-s + 8.19·27-s − 3.93·29-s + 4.89·31-s + 0.626·35-s − 2.49·37-s − 1.02·39-s + 8.92·41-s − 5.88·43-s − 2.18·45-s + 4.64·47-s − 4.26·49-s − 22.3·51-s − 1.62·53-s − 14.1·57-s − 2.93·59-s − 7.85·61-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 0.169·5-s − 0.624·7-s + 1.92·9-s − 0.0961·13-s − 0.289·15-s − 1.82·17-s − 1.09·19-s − 1.06·21-s − 1.64·23-s − 0.971·25-s + 1.57·27-s − 0.730·29-s + 0.879·31-s + 0.105·35-s − 0.410·37-s − 0.164·39-s + 1.39·41-s − 0.898·43-s − 0.325·45-s + 0.676·47-s − 0.609·49-s − 3.12·51-s − 0.223·53-s − 1.86·57-s − 0.382·59-s − 1.00·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 + 0.378T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.346T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 + 1.62T + 53T^{2} \) |
| 59 | \( 1 + 2.93T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 73 | \( 1 - 4.88T + 73T^{2} \) |
| 79 | \( 1 - 8.67T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 - 3.89T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.070911761714713272398019251228, −7.48312458892037263216261643289, −6.59049796380367080919931424151, −6.04514507270229453580599812737, −4.54410029875232956884696906217, −4.07511966356668294207778336480, −3.33973668060074907620429860904, −2.33153426611409961551783070705, −1.94150279371152359498374986712, 0,
1.94150279371152359498374986712, 2.33153426611409961551783070705, 3.33973668060074907620429860904, 4.07511966356668294207778336480, 4.54410029875232956884696906217, 6.04514507270229453580599812737, 6.59049796380367080919931424151, 7.48312458892037263216261643289, 8.070911761714713272398019251228
