| L(s) = 1 | − 1.79·2-s + 2.94·3-s + 1.21·4-s − 4.09·5-s − 5.27·6-s + 3.85·7-s + 1.41·8-s + 5.67·9-s + 7.33·10-s − 2.39·11-s + 3.57·12-s − 0.856·13-s − 6.90·14-s − 12.0·15-s − 4.95·16-s + 8.13·17-s − 10.1·18-s + 2.25·19-s − 4.96·20-s + 11.3·21-s + 4.29·22-s − 5.62·23-s + 4.15·24-s + 11.7·25-s + 1.53·26-s + 7.88·27-s + 4.67·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 1.70·3-s + 0.606·4-s − 1.83·5-s − 2.15·6-s + 1.45·7-s + 0.499·8-s + 1.89·9-s + 2.31·10-s − 0.722·11-s + 1.03·12-s − 0.237·13-s − 1.84·14-s − 3.11·15-s − 1.23·16-s + 1.97·17-s − 2.39·18-s + 0.517·19-s − 1.10·20-s + 2.47·21-s + 0.915·22-s − 1.17·23-s + 0.848·24-s + 2.35·25-s + 0.300·26-s + 1.51·27-s + 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136801082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136801082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 + 0.856T + 13T^{2} \) |
| 17 | \( 1 - 8.13T + 17T^{2} \) |
| 19 | \( 1 - 2.25T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 - 2.71T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 + 7.55T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 + 4.00T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 1.19T + 97T^{2} \) |
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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
−10.58985751275112233899914280134, −9.888567777004665665901967945462, −8.740588213843309087994413359608, −8.072664676754423988615271247911, −7.83269143775579323716827611520, −7.40946875431742238107613483653, −4.83856267514098596188128528687, −3.95925019708094393791032739258, −2.76216180150456172895315102543, −1.20661261939669521695090072129,
1.20661261939669521695090072129, 2.76216180150456172895315102543, 3.95925019708094393791032739258, 4.83856267514098596188128528687, 7.40946875431742238107613483653, 7.83269143775579323716827611520, 8.072664676754423988615271247911, 8.740588213843309087994413359608, 9.888567777004665665901967945462, 10.58985751275112233899914280134
