| L(s) = 1 | + 1.30·2-s − 0.285·4-s − 5-s + 4.28·7-s − 2.99·8-s − 1.30·10-s + 3.71·11-s + 1.17·13-s + 5.60·14-s − 3.34·16-s + 7.72·17-s − 0.216·19-s + 0.285·20-s + 4.86·22-s − 1.27·23-s + 25-s + 1.54·26-s − 1.22·28-s − 5.32·29-s − 3.94·31-s + 1.60·32-s + 10.1·34-s − 4.28·35-s + 0.356·37-s − 0.283·38-s + 2.99·40-s − 11.1·41-s + ⋯ |
| L(s) = 1 | + 0.925·2-s − 0.142·4-s − 0.447·5-s + 1.61·7-s − 1.05·8-s − 0.414·10-s + 1.11·11-s + 0.327·13-s + 1.49·14-s − 0.836·16-s + 1.87·17-s − 0.0496·19-s + 0.0639·20-s + 1.03·22-s − 0.266·23-s + 0.200·25-s + 0.302·26-s − 0.231·28-s − 0.989·29-s − 0.709·31-s + 0.283·32-s + 1.73·34-s − 0.723·35-s + 0.0586·37-s − 0.0459·38-s + 0.473·40-s − 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.310204575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.310204575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 7.72T + 17T^{2} \) |
| 19 | \( 1 + 0.216T + 19T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 0.356T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 - 8.60T + 67T^{2} \) |
| 71 | \( 1 + 7.59T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 97 | \( 1 - 7.73T + 97T^{2} \) |
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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
−8.360298009019187963673175581682, −7.77456192782299785586631076575, −6.96586473095799599033089812190, −5.91016465669130389290811062188, −5.34181033310566599589943505617, −4.69162687385132939659527700463, −3.81312536916455116433556153443, −3.46243032117449528132042646479, −1.98426297850536132153276409565, −0.979590348987273610100237029596,
0.979590348987273610100237029596, 1.98426297850536132153276409565, 3.46243032117449528132042646479, 3.81312536916455116433556153443, 4.69162687385132939659527700463, 5.34181033310566599589943505617, 5.91016465669130389290811062188, 6.96586473095799599033089812190, 7.77456192782299785586631076575, 8.360298009019187963673175581682
