| L(s) = 1 | − 2.18·2-s + 2.09·3-s + 2.76·4-s + 2.17·5-s − 4.57·6-s − 2.34·7-s − 1.67·8-s + 1.38·9-s − 4.75·10-s − 3.00·11-s + 5.79·12-s − 13-s + 5.12·14-s + 4.55·15-s − 1.87·16-s + 3.33·17-s − 3.01·18-s − 3.57·19-s + 6.03·20-s − 4.91·21-s + 6.55·22-s + 4.76·23-s − 3.51·24-s − 0.255·25-s + 2.18·26-s − 3.38·27-s − 6.49·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.20·3-s + 1.38·4-s + 0.974·5-s − 1.86·6-s − 0.887·7-s − 0.593·8-s + 0.460·9-s − 1.50·10-s − 0.905·11-s + 1.67·12-s − 0.277·13-s + 1.37·14-s + 1.17·15-s − 0.467·16-s + 0.809·17-s − 0.711·18-s − 0.819·19-s + 1.34·20-s − 1.07·21-s + 1.39·22-s + 0.993·23-s − 0.717·24-s − 0.0511·25-s + 0.428·26-s − 0.651·27-s − 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 463 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 8.52T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 4.03T + 67T^{2} \) |
| 71 | \( 1 - 5.45T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 - 5.08T + 79T^{2} \) |
| 83 | \( 1 + 7.60T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−8.105539288019512231287681917125, −7.20171238125350660833321296030, −6.70215764807009163970909365465, −5.80113378093131181844564305997, −4.93467750123990835609146379506, −3.62941680251840915180162632303, −2.73916491061402563809822687119, −2.33046636731958845475576766047, −1.34799987872016926456753689707, 0,
1.34799987872016926456753689707, 2.33046636731958845475576766047, 2.73916491061402563809822687119, 3.62941680251840915180162632303, 4.93467750123990835609146379506, 5.80113378093131181844564305997, 6.70215764807009163970909365465, 7.20171238125350660833321296030, 8.105539288019512231287681917125
