| L(s) = 1 | − 3.60·3-s − 19.7·5-s − 7·7-s − 14.0·9-s + 11·11-s + 0.0182·13-s + 71.1·15-s + 41.9·17-s + 20.9·19-s + 25.2·21-s − 47.5·23-s + 265.·25-s + 147.·27-s + 141.·29-s + 1.39·31-s − 39.6·33-s + 138.·35-s − 67.5·37-s − 0.0656·39-s + 211.·41-s + 101.·43-s + 276.·45-s − 548.·47-s + 49·49-s − 151.·51-s + 701.·53-s − 217.·55-s + ⋯ |
| L(s) = 1 | − 0.693·3-s − 1.76·5-s − 0.377·7-s − 0.519·9-s + 0.301·11-s + 0.000388·13-s + 1.22·15-s + 0.598·17-s + 0.253·19-s + 0.262·21-s − 0.430·23-s + 2.12·25-s + 1.05·27-s + 0.903·29-s + 0.00809·31-s − 0.209·33-s + 0.667·35-s − 0.300·37-s − 0.000269·39-s + 0.807·41-s + 0.360·43-s + 0.917·45-s − 1.70·47-s + 0.142·49-s − 0.415·51-s + 1.81·53-s − 0.532·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 3.60T + 27T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 13 | \( 1 - 0.0182T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 47.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 1.39T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 701.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 66.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 175.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 168.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 163.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 119.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 86.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 507.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−8.718208247381990818541459756279, −8.122376802123531773144915573683, −7.28312713994604720139072207485, −6.50448400110678932381187925385, −5.53231917221928221409785985297, −4.56621241759995096655779570342, −3.71027064984486985497784032491, −2.86655449367963239661397147184, −0.918259867719945211447512254193, 0,
0.918259867719945211447512254193, 2.86655449367963239661397147184, 3.71027064984486985497784032491, 4.56621241759995096655779570342, 5.53231917221928221409785985297, 6.50448400110678932381187925385, 7.28312713994604720139072207485, 8.122376802123531773144915573683, 8.718208247381990818541459756279
