| L(s) = 1 | − 0.226·2-s − 1.94·4-s − 3.14·5-s − 2.84·7-s + 0.894·8-s + 0.712·10-s − 11-s + 3.54·13-s + 0.644·14-s + 3.69·16-s − 6.41·17-s + 0.806·19-s + 6.13·20-s + 0.226·22-s − 4.07·23-s + 4.90·25-s − 0.801·26-s + 5.54·28-s + 0.826·29-s + 0.425·31-s − 2.62·32-s + 1.45·34-s + 8.95·35-s + 5.03·37-s − 0.182·38-s − 2.81·40-s + 9.88·41-s + ⋯ |
| L(s) = 1 | − 0.160·2-s − 0.974·4-s − 1.40·5-s − 1.07·7-s + 0.316·8-s + 0.225·10-s − 0.301·11-s + 0.981·13-s + 0.172·14-s + 0.923·16-s − 1.55·17-s + 0.185·19-s + 1.37·20-s + 0.0482·22-s − 0.849·23-s + 0.981·25-s − 0.157·26-s + 1.04·28-s + 0.153·29-s + 0.0765·31-s − 0.464·32-s + 0.248·34-s + 1.51·35-s + 0.827·37-s − 0.0296·38-s − 0.444·40-s + 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.226T + 2T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 - 0.806T + 19T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 - 0.826T + 29T^{2} \) |
| 31 | \( 1 - 0.425T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 0.550T + 53T^{2} \) |
| 59 | \( 1 - 4.11T + 59T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.68T + 73T^{2} \) |
| 79 | \( 1 + 9.00T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 9.63T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.945322384953830090759572494524, −7.10488384396574499452008341174, −6.34396229308377514612143724588, −5.61266651467711652528140693927, −4.49386391799348360038930315750, −4.05729558438461730272043233495, −3.50491726893910916197555640627, −2.49203204831397496026878074587, −0.841480105872876958148681869176, 0,
0.841480105872876958148681869176, 2.49203204831397496026878074587, 3.50491726893910916197555640627, 4.05729558438461730272043233495, 4.49386391799348360038930315750, 5.61266651467711652528140693927, 6.34396229308377514612143724588, 7.10488384396574499452008341174, 7.945322384953830090759572494524
