| L(s) = 1 | − 2.77·2-s − 0.494·3-s + 5.68·4-s + 0.905·5-s + 1.36·6-s + 1.77·7-s − 10.2·8-s − 2.75·9-s − 2.51·10-s − 5.29·11-s − 2.80·12-s + 3.37·13-s − 4.92·14-s − 0.447·15-s + 16.9·16-s + 5.30·17-s + 7.63·18-s + 8.13·19-s + 5.14·20-s − 0.878·21-s + 14.6·22-s + 8.87·23-s + 5.04·24-s − 4.17·25-s − 9.36·26-s + 2.84·27-s + 10.1·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.285·3-s + 2.84·4-s + 0.405·5-s + 0.559·6-s + 0.672·7-s − 3.61·8-s − 0.918·9-s − 0.793·10-s − 1.59·11-s − 0.810·12-s + 0.936·13-s − 1.31·14-s − 0.115·15-s + 4.23·16-s + 1.28·17-s + 1.80·18-s + 1.86·19-s + 1.15·20-s − 0.191·21-s + 3.13·22-s + 1.85·23-s + 1.03·24-s − 0.835·25-s − 1.83·26-s + 0.547·27-s + 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7950357510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7950357510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4019 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 0.494T + 3T^{2} \) |
| 5 | \( 1 - 0.905T + 5T^{2} \) |
| 7 | \( 1 - 1.77T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 8.13T + 19T^{2} \) |
| 23 | \( 1 - 8.87T + 23T^{2} \) |
| 29 | \( 1 - 0.398T + 29T^{2} \) |
| 31 | \( 1 - 3.20T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 + 0.418T + 43T^{2} \) |
| 47 | \( 1 - 5.75T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 - 7.32T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−8.317654707166874515539925372326, −7.924290837859527446487596692314, −7.41218729018204637445244552714, −6.41930109259227257651745581171, −5.53337046864297586104429820614, −5.29350190001119784025794347053, −3.12820162702863573217114584619, −2.79444809796770877966477322199, −1.50257311288251481771727303399, −0.74236593414903787639195605296,
0.74236593414903787639195605296, 1.50257311288251481771727303399, 2.79444809796770877966477322199, 3.12820162702863573217114584619, 5.29350190001119784025794347053, 5.53337046864297586104429820614, 6.41930109259227257651745581171, 7.41218729018204637445244552714, 7.924290837859527446487596692314, 8.317654707166874515539925372326
