| L(s) = 1 | − 1.79·2-s + 3-s + 1.23·4-s + 3.18·5-s − 1.79·6-s + 1.45·7-s + 1.38·8-s + 9-s − 5.72·10-s + 5.12·11-s + 1.23·12-s − 13-s − 2.61·14-s + 3.18·15-s − 4.94·16-s + 2.85·17-s − 1.79·18-s − 5.56·19-s + 3.92·20-s + 1.45·21-s − 9.21·22-s − 4.19·23-s + 1.38·24-s + 5.15·25-s + 1.79·26-s + 27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.615·4-s + 1.42·5-s − 0.733·6-s + 0.549·7-s + 0.488·8-s + 0.333·9-s − 1.81·10-s + 1.54·11-s + 0.355·12-s − 0.277·13-s − 0.698·14-s + 0.822·15-s − 1.23·16-s + 0.693·17-s − 0.423·18-s − 1.27·19-s + 0.877·20-s + 0.317·21-s − 1.96·22-s − 0.875·23-s + 0.281·24-s + 1.03·25-s + 0.352·26-s + 0.192·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917736341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917736341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 0.718T + 29T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 3.88T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 + 0.802T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 + 0.798T + 53T^{2} \) |
| 59 | \( 1 - 7.71T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 - 1.20T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 4.93T + 73T^{2} \) |
| 79 | \( 1 + 0.0933T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 6.21T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
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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
−8.515794075616339855836913176954, −8.066574289361983815914808495348, −7.09507826034380821290675851918, −6.45821635390499333724842160232, −5.69137154481381770153707249589, −4.59253950179446563650232181216, −3.85197879704627145468609123062, −2.37351748646533609154970858174, −1.82582307486390303897139736076, −1.01872847750349441832069390395,
1.01872847750349441832069390395, 1.82582307486390303897139736076, 2.37351748646533609154970858174, 3.85197879704627145468609123062, 4.59253950179446563650232181216, 5.69137154481381770153707249589, 6.45821635390499333724842160232, 7.09507826034380821290675851918, 8.066574289361983815914808495348, 8.515794075616339855836913176954
