| L(s) = 1 | + 2-s − 0.0253·3-s + 4-s + 0.720·5-s − 0.0253·6-s − 3.77·7-s + 8-s − 2.99·9-s + 0.720·10-s + 1.69·11-s − 0.0253·12-s + 3.77·13-s − 3.77·14-s − 0.0182·15-s + 16-s − 0.226·17-s − 2.99·18-s + 4.47·19-s + 0.720·20-s + 0.0956·21-s + 1.69·22-s − 8.44·23-s − 0.0253·24-s − 4.48·25-s + 3.77·26-s + 0.151·27-s − 3.77·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0146·3-s + 0.5·4-s + 0.322·5-s − 0.0103·6-s − 1.42·7-s + 0.353·8-s − 0.999·9-s + 0.227·10-s + 0.511·11-s − 0.00730·12-s + 1.04·13-s − 1.00·14-s − 0.00470·15-s + 0.250·16-s − 0.0548·17-s − 0.706·18-s + 1.02·19-s + 0.161·20-s + 0.0208·21-s + 0.361·22-s − 1.76·23-s − 0.00516·24-s − 0.896·25-s + 0.740·26-s + 0.0292·27-s − 0.713·28-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 | \( 1 + 0.0253T + 3T^{2} \) |
| 5 | \( 1 - 0.720T + 5T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 0.226T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 8.44T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 - 0.438T + 37T^{2} \) |
| 41 | \( 1 - 0.946T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 - 0.852T + 53T^{2} \) |
| 59 | \( 1 - 5.08T + 59T^{2} \) |
| 61 | \( 1 - 0.0834T + 61T^{2} \) |
| 67 | \( 1 + 5.33T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 - 3.16T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.056424944866577974145175617762, −7.13743712086504051824173523959, −6.26583123716764585736635207836, −5.97046691539413662228377868753, −5.32637903602033481959935094330, −3.94592852412154633072637773670, −3.54814504789280148980768599558, −2.72534522364343127093750695747, −1.61927856138549721929007356687, 0,
1.61927856138549721929007356687, 2.72534522364343127093750695747, 3.54814504789280148980768599558, 3.94592852412154633072637773670, 5.32637903602033481959935094330, 5.97046691539413662228377868753, 6.26583123716764585736635207836, 7.13743712086504051824173523959, 8.056424944866577974145175617762
