L(s) = 1 | − 2-s + 2.86·3-s + 4-s + 0.449·5-s − 2.86·6-s − 3.16·7-s − 8-s + 5.20·9-s − 0.449·10-s + 3.87·11-s + 2.86·12-s + 1.09·13-s + 3.16·14-s + 1.28·15-s + 16-s − 2.07·17-s − 5.20·18-s + 1.06·19-s + 0.449·20-s − 9.07·21-s − 3.87·22-s + 6.37·23-s − 2.86·24-s − 4.79·25-s − 1.09·26-s + 6.33·27-s − 3.16·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.65·3-s + 0.5·4-s + 0.200·5-s − 1.16·6-s − 1.19·7-s − 0.353·8-s + 1.73·9-s − 0.142·10-s + 1.16·11-s + 0.827·12-s + 0.304·13-s + 0.846·14-s + 0.332·15-s + 0.250·16-s − 0.502·17-s − 1.22·18-s + 0.243·19-s + 0.100·20-s − 1.98·21-s − 0.825·22-s + 1.32·23-s − 0.584·24-s − 0.959·25-s − 0.215·26-s + 1.21·27-s − 0.598·28-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
\(L(1)\) |
\(\approx\) |
\(2.611560124\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.611560124\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 - 0.449T + 5T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 5.99T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 - 2.71T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 - 9.76T + 79T^{2} \) |
| 83 | \( 1 + 0.392T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 + 0.655T + 97T^{2} \) |
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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
−8.631516463291448198820031479520, −7.908535829881682171014093247340, −7.01765813652297614194292543599, −6.65847914423621239454445767924, −5.71826409769132499184667914258, −4.24447080551151569151206864882, −3.58209214975993986437553424330, −2.87545776801423990625830203514, −2.08764821324553398473814598810, −0.981103209148228820394182859958,
0.981103209148228820394182859958, 2.08764821324553398473814598810, 2.87545776801423990625830203514, 3.58209214975993986437553424330, 4.24447080551151569151206864882, 5.71826409769132499184667914258, 6.65847914423621239454445767924, 7.01765813652297614194292543599, 7.908535829881682171014093247340, 8.631516463291448198820031479520