L(s) = 1 | + 2-s − 2.46·3-s + 4-s − 2.36·5-s − 2.46·6-s + 2.92·7-s + 8-s + 3.06·9-s − 2.36·10-s + 0.754·11-s − 2.46·12-s − 4.60·13-s + 2.92·14-s + 5.83·15-s + 16-s − 3.30·17-s + 3.06·18-s + 4.18·19-s − 2.36·20-s − 7.20·21-s + 0.754·22-s + 6.30·23-s − 2.46·24-s + 0.616·25-s − 4.60·26-s − 0.153·27-s + 2.92·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s − 1.05·5-s − 1.00·6-s + 1.10·7-s + 0.353·8-s + 1.02·9-s − 0.749·10-s + 0.227·11-s − 0.710·12-s − 1.27·13-s + 0.782·14-s + 1.50·15-s + 0.250·16-s − 0.802·17-s + 0.721·18-s + 0.959·19-s − 0.529·20-s − 1.57·21-s + 0.160·22-s + 1.31·23-s − 0.502·24-s + 0.123·25-s − 0.903·26-s − 0.0296·27-s + 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 - 0.754T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 + 3.30T + 17T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 + 0.964T + 37T^{2} \) |
| 41 | \( 1 + 3.77T + 41T^{2} \) |
| 43 | \( 1 - 8.35T + 43T^{2} \) |
| 47 | \( 1 + 5.98T + 47T^{2} \) |
| 53 | \( 1 - 8.46T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 - 6.86T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 + 9.97T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 - 3.64T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−7.70833718534221078051183218355, −7.21270739850267223894975993125, −6.66955382565883772388867962647, −5.38472074697179887474604168518, −5.24039015755987471475972666836, −4.46993387910352651998735385256, −3.77167729260524953167236789170, −2.52780569992730485441805463972, −1.27176566276286482192953811298, 0,
1.27176566276286482192953811298, 2.52780569992730485441805463972, 3.77167729260524953167236789170, 4.46993387910352651998735385256, 5.24039015755987471475972666836, 5.38472074697179887474604168518, 6.66955382565883772388867962647, 7.21270739850267223894975993125, 7.70833718534221078051183218355