| L(s) = 1 | + 2-s − 3-s + 4-s + 0.721·5-s − 6-s + 8-s + 9-s + 0.721·10-s + 2.36·11-s − 12-s − 1.15·13-s − 0.721·15-s + 16-s − 1.17·17-s + 18-s − 4.75·19-s + 0.721·20-s + 2.36·22-s + 23-s − 24-s − 4.47·25-s − 1.15·26-s − 27-s + 2.49·29-s − 0.721·30-s − 4.02·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.322·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.228·10-s + 0.712·11-s − 0.288·12-s − 0.320·13-s − 0.186·15-s + 0.250·16-s − 0.284·17-s + 0.235·18-s − 1.09·19-s + 0.161·20-s + 0.503·22-s + 0.208·23-s − 0.204·24-s − 0.895·25-s − 0.226·26-s − 0.192·27-s + 0.464·29-s − 0.131·30-s − 0.723·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.721T + 5T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 4.84T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 + 5.68T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 - 5.70T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.33061209508332313376408328858, −6.65885946321765049311806304566, −6.31126593698340515189463223008, −5.37682484870394933770187226465, −4.90196373141382403959804599444, −4.02810615121825606368263199406, −3.39814649138863554282638123116, −2.19696434816238819605873804993, −1.53626167210685166070310844860, 0,
1.53626167210685166070310844860, 2.19696434816238819605873804993, 3.39814649138863554282638123116, 4.02810615121825606368263199406, 4.90196373141382403959804599444, 5.37682484870394933770187226465, 6.31126593698340515189463223008, 6.65885946321765049311806304566, 7.33061209508332313376408328858
