| L(s) = 1 | − 0.230·2-s − 1.94·4-s + 2.49·5-s − 0.810·7-s + 0.907·8-s − 0.573·10-s + 0.0355·11-s − 5.55·13-s + 0.186·14-s + 3.68·16-s − 4.02·17-s − 19-s − 4.85·20-s − 0.00817·22-s + 2.50·23-s + 1.20·25-s + 1.27·26-s + 1.57·28-s − 4.83·29-s − 2.28·31-s − 2.66·32-s + 0.925·34-s − 2.01·35-s + 7.13·37-s + 0.230·38-s + 2.26·40-s − 3.34·41-s + ⋯ |
| L(s) = 1 | − 0.162·2-s − 0.973·4-s + 1.11·5-s − 0.306·7-s + 0.320·8-s − 0.181·10-s + 0.0107·11-s − 1.54·13-s + 0.0498·14-s + 0.921·16-s − 0.976·17-s − 0.229·19-s − 1.08·20-s − 0.00174·22-s + 0.521·23-s + 0.241·25-s + 0.250·26-s + 0.298·28-s − 0.896·29-s − 0.410·31-s − 0.470·32-s + 0.158·34-s − 0.341·35-s + 1.17·37-s + 0.0373·38-s + 0.357·40-s − 0.522·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131697988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131697988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 0.230T + 2T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 0.810T + 7T^{2} \) |
| 11 | \( 1 - 0.0355T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 - 7.13T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 + 0.765T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 - 7.91T + 67T^{2} \) |
| 71 | \( 1 - 2.89T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 0.389T + 97T^{2} \) |
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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
−7.888787521498451393565296111879, −7.11272738499261654586120278391, −6.46416289768878081555469915870, −5.59903329002572261512175523408, −5.08655007916269915883541864110, −4.43701099823933043496865697893, −3.55101952606781168953585167902, −2.49617441024284330893584457804, −1.85900929887824730819818980238, −0.52746470705506317215730050743,
0.52746470705506317215730050743, 1.85900929887824730819818980238, 2.49617441024284330893584457804, 3.55101952606781168953585167902, 4.43701099823933043496865697893, 5.08655007916269915883541864110, 5.59903329002572261512175523408, 6.46416289768878081555469915870, 7.11272738499261654586120278391, 7.888787521498451393565296111879
