| L(s) = 1 | − 2.56·2-s − 0.512·3-s + 4.60·4-s − 4.01·5-s + 1.31·6-s + 0.174·7-s − 6.69·8-s − 2.73·9-s + 10.3·10-s − 2.58·11-s − 2.36·12-s + 0.398·13-s − 0.449·14-s + 2.05·15-s + 7.99·16-s + 2.25·17-s + 7.03·18-s + 4.68·19-s − 18.4·20-s − 0.0895·21-s + 6.65·22-s − 6.94·23-s + 3.43·24-s + 11.0·25-s − 1.02·26-s + 2.94·27-s + 0.804·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.295·3-s + 2.30·4-s − 1.79·5-s + 0.537·6-s + 0.0660·7-s − 2.36·8-s − 0.912·9-s + 3.26·10-s − 0.780·11-s − 0.681·12-s + 0.110·13-s − 0.120·14-s + 0.530·15-s + 1.99·16-s + 0.547·17-s + 1.65·18-s + 1.07·19-s − 4.13·20-s − 0.0195·21-s + 1.41·22-s − 1.44·23-s + 0.700·24-s + 2.21·25-s − 0.201·26-s + 0.565·27-s + 0.152·28-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
| \(L(1)\) |
\(\approx\) |
\(0.07913326432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07913326432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 0.512T + 3T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 7 | \( 1 - 0.174T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 0.398T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 - 0.630T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 0.895T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 + 5.63T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 0.160T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 + 4.74T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−8.298321039363364689168114086724, −7.900611095810507049011956545335, −7.44136790927635943034660254452, −6.61578621773474099308670072786, −5.69866777618084841745258743998, −4.73512405140416793358118394377, −3.43374358566989160972212269965, −2.90716426435038381032054137745, −1.50695865036746605952357561964, −0.21731875526645175043336423202,
0.21731875526645175043336423202, 1.50695865036746605952357561964, 2.90716426435038381032054137745, 3.43374358566989160972212269965, 4.73512405140416793358118394377, 5.69866777618084841745258743998, 6.61578621773474099308670072786, 7.44136790927635943034660254452, 7.900611095810507049011956545335, 8.298321039363364689168114086724
