| L(s) = 1 | − 2.79·2-s − 3.39·3-s + 5.79·4-s + 3.31·5-s + 9.47·6-s − 0.0588·7-s − 10.5·8-s + 8.51·9-s − 9.24·10-s + 3.18·11-s − 19.6·12-s + 1.79·13-s + 0.164·14-s − 11.2·15-s + 17.9·16-s + 6.19·17-s − 23.7·18-s + 4.49·19-s + 19.1·20-s + 0.199·21-s − 8.90·22-s − 5.64·23-s + 35.9·24-s + 5.96·25-s − 5.01·26-s − 18.7·27-s − 0.340·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 1.95·3-s + 2.89·4-s + 1.48·5-s + 3.86·6-s − 0.0222·7-s − 3.74·8-s + 2.83·9-s − 2.92·10-s + 0.961·11-s − 5.67·12-s + 0.498·13-s + 0.0438·14-s − 2.90·15-s + 4.49·16-s + 1.50·17-s − 5.60·18-s + 1.03·19-s + 4.28·20-s + 0.0435·21-s − 1.89·22-s − 1.17·23-s + 7.33·24-s + 1.19·25-s − 0.983·26-s − 3.60·27-s − 0.0643·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7255491803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7255491803\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 0.0588T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 + 5.64T + 23T^{2} \) |
| 29 | \( 1 - 0.468T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 1.69T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 8.29T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 0.0659T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 0.766T + 89T^{2} \) |
| 97 | \( 1 - 8.17T + 97T^{2} \) |
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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
−8.635647117738443597921816583676, −7.54420288639112499835976021883, −7.00060133596889855465779121238, −6.21445993803239180330634081714, −5.87370044726338861774394166259, −5.36429949680425568446591112854, −3.73281362445833063035618986090, −2.22029934343133158854029081415, −1.26026471867218775186456512474, −0.893408609896576976289671080877,
0.893408609896576976289671080877, 1.26026471867218775186456512474, 2.22029934343133158854029081415, 3.73281362445833063035618986090, 5.36429949680425568446591112854, 5.87370044726338861774394166259, 6.21445993803239180330634081714, 7.00060133596889855465779121238, 7.54420288639112499835976021883, 8.635647117738443597921816583676
