| L(s) = 1 | − 2-s + 3.04·3-s + 4-s + 3.23·5-s − 3.04·6-s + 4.92·7-s − 8-s + 6.29·9-s − 3.23·10-s − 2.68·11-s + 3.04·12-s − 0.219·13-s − 4.92·14-s + 9.86·15-s + 16-s + 0.108·17-s − 6.29·18-s − 19-s + 3.23·20-s + 15.0·21-s + 2.68·22-s − 4.51·23-s − 3.04·24-s + 5.46·25-s + 0.219·26-s + 10.0·27-s + 4.92·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s + 1.44·5-s − 1.24·6-s + 1.86·7-s − 0.353·8-s + 2.09·9-s − 1.02·10-s − 0.808·11-s + 0.879·12-s − 0.0609·13-s − 1.31·14-s + 2.54·15-s + 0.250·16-s + 0.0263·17-s − 1.48·18-s − 0.229·19-s + 0.723·20-s + 3.27·21-s + 0.571·22-s − 0.942·23-s − 0.622·24-s + 1.09·25-s + 0.0430·26-s + 1.93·27-s + 0.930·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.003872330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.003872330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 + 0.219T + 13T^{2} \) |
| 17 | \( 1 - 0.108T + 17T^{2} \) |
| 23 | \( 1 + 4.51T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 7.69T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 8.54T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 4.90T + 71T^{2} \) |
| 73 | \( 1 + 6.09T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 + 8.15T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−8.066874035438803979354520913763, −7.58984273783952305409944137628, −6.65673021637686548546266638623, −5.81163910108126841488494539417, −4.92526726031964860114381970045, −4.33274996369455542780247337110, −3.07069408929589806510582529177, −2.37782747738142509451573114008, −1.88587534431074589772030730990, −1.29175852818707078165259913413,
1.29175852818707078165259913413, 1.88587534431074589772030730990, 2.37782747738142509451573114008, 3.07069408929589806510582529177, 4.33274996369455542780247337110, 4.92526726031964860114381970045, 5.81163910108126841488494539417, 6.65673021637686548546266638623, 7.58984273783952305409944137628, 8.066874035438803979354520913763
