| L(s) = 1 | − 2-s − 1.68·3-s − 4-s − 4.18·5-s + 1.68·6-s + 1.68·7-s + 3·8-s − 0.173·9-s + 4.18·10-s + 1.68·12-s + 3.87·13-s − 1.68·14-s + 7.04·15-s − 16-s − 17-s + 0.173·18-s − 0.318·19-s + 4.18·20-s − 2.82·21-s + 5.87·23-s − 5.04·24-s + 12.5·25-s − 3.87·26-s + 5.33·27-s − 1.68·28-s − 7.55·29-s − 7.04·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.970·3-s − 0.5·4-s − 1.87·5-s + 0.686·6-s + 0.635·7-s + 1.06·8-s − 0.0577·9-s + 1.32·10-s + 0.485·12-s + 1.07·13-s − 0.449·14-s + 1.81·15-s − 0.250·16-s − 0.242·17-s + 0.0408·18-s − 0.0731·19-s + 0.936·20-s − 0.616·21-s + 1.22·23-s − 1.02·24-s + 2.51·25-s − 0.759·26-s + 1.02·27-s − 0.317·28-s − 1.40·29-s − 1.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 19 | \( 1 + 0.318T + 19T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 + 6.50T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 + 6.88T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 - 3.36T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 1.01T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−9.334182989338245451623834581613, −8.674507229020757055811726723855, −7.87698024028393778671239939590, −7.40702452866015130371296352644, −6.17601850343760958258734798693, −4.98896944906530395292962361871, −4.36838788236790300490124711263, −3.40307635245879645947400480037, −1.12796201658711298542593066247, 0,
1.12796201658711298542593066247, 3.40307635245879645947400480037, 4.36838788236790300490124711263, 4.98896944906530395292962361871, 6.17601850343760958258734798693, 7.40702452866015130371296352644, 7.87698024028393778671239939590, 8.674507229020757055811726723855, 9.334182989338245451623834581613
