| L(s) = 1 | − 0.574·2-s + 0.252·3-s − 1.66·4-s − 5-s − 0.145·6-s + 2.10·8-s − 2.93·9-s + 0.574·10-s − 0.276·11-s − 0.422·12-s − 2.81·13-s − 0.252·15-s + 2.12·16-s − 1.85·17-s + 1.68·18-s + 5.50·19-s + 1.66·20-s + 0.159·22-s − 0.269·23-s + 0.533·24-s + 25-s + 1.61·26-s − 1.50·27-s + 29-s + 0.145·30-s − 0.508·31-s − 5.44·32-s + ⋯ |
| L(s) = 1 | − 0.406·2-s + 0.145·3-s − 0.834·4-s − 0.447·5-s − 0.0593·6-s + 0.745·8-s − 0.978·9-s + 0.181·10-s − 0.0834·11-s − 0.121·12-s − 0.779·13-s − 0.0652·15-s + 0.531·16-s − 0.449·17-s + 0.397·18-s + 1.26·19-s + 0.373·20-s + 0.0339·22-s − 0.0562·23-s + 0.108·24-s + 0.200·25-s + 0.316·26-s − 0.288·27-s + 0.185·29-s + 0.0265·30-s − 0.0914·31-s − 0.961·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.574T + 2T^{2} \) |
| 3 | \( 1 - 0.252T + 3T^{2} \) |
| 11 | \( 1 + 0.276T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 + 0.269T + 23T^{2} \) |
| 31 | \( 1 + 0.508T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 + 0.703T + 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 - 5.83T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 1.13T + 83T^{2} \) |
| 89 | \( 1 - 6.72T + 89T^{2} \) |
| 97 | \( 1 - 2.00T + 97T^{2} \) |
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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
−7.80304417895872479370222098519, −7.13433134316734157436359675625, −6.16407722606784150387544939654, −5.28818912329901626068195967154, −4.83248120779836385318297740297, −3.94219081574390741287339241983, −3.17962872377194462214262437439, −2.32184418283286602177675087733, −0.980377263654519216301931863761, 0,
0.980377263654519216301931863761, 2.32184418283286602177675087733, 3.17962872377194462214262437439, 3.94219081574390741287339241983, 4.83248120779836385318297740297, 5.28818912329901626068195967154, 6.16407722606784150387544939654, 7.13433134316734157436359675625, 7.80304417895872479370222098519
