| L(s) = 1 | − 2.61·2-s + 0.347·3-s + 4.86·4-s + 3.06·5-s − 0.909·6-s − 2.40·7-s − 7.49·8-s − 2.87·9-s − 8.02·10-s − 5.26·11-s + 1.68·12-s − 3.23·13-s + 6.29·14-s + 1.06·15-s + 9.91·16-s − 1.08·17-s + 7.54·18-s + 0.530·19-s + 14.9·20-s − 0.834·21-s + 13.7·22-s − 6.75·23-s − 2.60·24-s + 4.39·25-s + 8.48·26-s − 2.04·27-s − 11.6·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.200·3-s + 2.43·4-s + 1.37·5-s − 0.371·6-s − 0.908·7-s − 2.65·8-s − 0.959·9-s − 2.53·10-s − 1.58·11-s + 0.487·12-s − 0.898·13-s + 1.68·14-s + 0.274·15-s + 2.47·16-s − 0.263·17-s + 1.77·18-s + 0.121·19-s + 3.33·20-s − 0.182·21-s + 2.94·22-s − 1.40·23-s − 0.531·24-s + 0.879·25-s + 1.66·26-s − 0.393·27-s − 2.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4439682301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4439682301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 + 5.26T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 - 0.530T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 8.35T + 29T^{2} \) |
| 31 | \( 1 + 8.45T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 3.29T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.99T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 - 1.95T + 71T^{2} \) |
| 73 | \( 1 + 0.967T + 73T^{2} \) |
| 79 | \( 1 + 5.71T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−8.625666451095001097948348387360, −7.80994414110278431523161090200, −7.32434667674795618012081798186, −6.17002772664636760539422201088, −6.02055508481862544135786742148, −5.02196151288830211370278804693, −3.17354330173243520260171073690, −2.43339203762820532673733402213, −2.08397964750334542709929986886, −0.45585294354716450237174391159,
0.45585294354716450237174391159, 2.08397964750334542709929986886, 2.43339203762820532673733402213, 3.17354330173243520260171073690, 5.02196151288830211370278804693, 6.02055508481862544135786742148, 6.17002772664636760539422201088, 7.32434667674795618012081798186, 7.80994414110278431523161090200, 8.625666451095001097948348387360
