| L(s) = 1 | − 2-s + 0.00791·3-s + 4-s − 2.05·5-s − 0.00791·6-s − 2.16·7-s − 8-s − 2.99·9-s + 2.05·10-s + 3.51·11-s + 0.00791·12-s + 0.954·13-s + 2.16·14-s − 0.0162·15-s + 16-s + 1.50·17-s + 2.99·18-s + 5.51·19-s − 2.05·20-s − 0.0171·21-s − 3.51·22-s + 2.04·23-s − 0.00791·24-s − 0.786·25-s − 0.954·26-s − 0.0474·27-s − 2.16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.00456·3-s + 0.5·4-s − 0.917·5-s − 0.00323·6-s − 0.817·7-s − 0.353·8-s − 0.999·9-s + 0.649·10-s + 1.05·11-s + 0.00228·12-s + 0.264·13-s + 0.578·14-s − 0.00419·15-s + 0.250·16-s + 0.364·17-s + 0.707·18-s + 1.26·19-s − 0.458·20-s − 0.00373·21-s − 0.749·22-s + 0.426·23-s − 0.00161·24-s − 0.157·25-s − 0.187·26-s − 0.00913·27-s − 0.408·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 0.00791T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 - 0.954T + 13T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 - 9.55T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 6.16T + 61T^{2} \) |
| 67 | \( 1 - 6.69T + 67T^{2} \) |
| 71 | \( 1 - 6.36T + 71T^{2} \) |
| 73 | \( 1 - 9.38T + 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−8.163693148149935118184223634352, −7.30584127264807514633783980176, −6.91429399739104281925965516744, −5.87703004111422311933928205435, −5.34976251373255940601901395550, −3.73408804006823330177322793422, −3.61572212731269147671189756195, −2.48999923883054034621715778937, −1.11854485969210514583368251903, 0,
1.11854485969210514583368251903, 2.48999923883054034621715778937, 3.61572212731269147671189756195, 3.73408804006823330177322793422, 5.34976251373255940601901395550, 5.87703004111422311933928205435, 6.91429399739104281925965516744, 7.30584127264807514633783980176, 8.163693148149935118184223634352
