| L(s) = 1 | − 1.95·2-s − 3-s + 1.81·4-s − 1.54·5-s + 1.95·6-s + 2.71·7-s + 0.369·8-s + 9-s + 3.01·10-s − 1.66·11-s − 1.81·12-s − 3.33·13-s − 5.29·14-s + 1.54·15-s − 4.34·16-s − 17-s − 1.95·18-s − 1.70·19-s − 2.79·20-s − 2.71·21-s + 3.24·22-s + 4.45·23-s − 0.369·24-s − 2.61·25-s + 6.50·26-s − 27-s + 4.91·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.905·4-s − 0.691·5-s + 0.796·6-s + 1.02·7-s + 0.130·8-s + 0.333·9-s + 0.954·10-s − 0.501·11-s − 0.522·12-s − 0.924·13-s − 1.41·14-s + 0.399·15-s − 1.08·16-s − 0.242·17-s − 0.460·18-s − 0.391·19-s − 0.625·20-s − 0.592·21-s + 0.692·22-s + 0.928·23-s − 0.0753·24-s − 0.522·25-s + 1.27·26-s − 0.192·27-s + 0.928·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 - 9.63T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 6.79T + 53T^{2} \) |
| 59 | \( 1 + 2.46T + 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 + 0.665T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 + 0.675T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 6.98T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 5.15T + 97T^{2} \) |
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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
−7.53303788608162627455765992114, −7.23256137399156536414272196967, −6.34992212838996481972704925514, −5.30765093467786843882424359126, −4.69250923060694742172013350512, −4.14822629827437843536274342204, −2.76350508589388000869138042956, −1.91149586353218468849855200902, −0.932199538729390722811300212816, 0,
0.932199538729390722811300212816, 1.91149586353218468849855200902, 2.76350508589388000869138042956, 4.14822629827437843536274342204, 4.69250923060694742172013350512, 5.30765093467786843882424359126, 6.34992212838996481972704925514, 7.23256137399156536414272196967, 7.53303788608162627455765992114
