| L(s) = 1 | − 2.50·2-s + 3.16·3-s + 4.25·4-s − 4.26·5-s − 7.90·6-s − 1.67·7-s − 5.62·8-s + 7.00·9-s + 10.6·10-s + 11-s + 13.4·12-s − 1.86·13-s + 4.19·14-s − 13.4·15-s + 5.56·16-s + 0.973·17-s − 17.5·18-s + 1.53·19-s − 18.1·20-s − 5.30·21-s − 2.50·22-s − 4.14·23-s − 17.8·24-s + 13.1·25-s + 4.66·26-s + 12.6·27-s − 7.12·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.82·3-s + 2.12·4-s − 1.90·5-s − 3.22·6-s − 0.633·7-s − 1.98·8-s + 2.33·9-s + 3.37·10-s + 0.301·11-s + 3.88·12-s − 0.517·13-s + 1.11·14-s − 3.48·15-s + 1.39·16-s + 0.236·17-s − 4.12·18-s + 0.351·19-s − 4.05·20-s − 1.15·21-s − 0.533·22-s − 0.863·23-s − 3.63·24-s + 2.63·25-s + 0.915·26-s + 2.43·27-s − 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 0.973T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 9.99T + 61T^{2} \) |
| 67 | \( 1 + 0.0984T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 - 7.11T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.846429802662856184058682086077, −8.429855211758968290783980204213, −7.74193531672825538760351673707, −7.36237339276501362948391011349, −6.62656069454880153032871613249, −4.44184968707833639973263724465, −3.46194679566087106026315987743, −2.89966910159384699744412060908, −1.57880671453704689187330265211, 0,
1.57880671453704689187330265211, 2.89966910159384699744412060908, 3.46194679566087106026315987743, 4.44184968707833639973263724465, 6.62656069454880153032871613249, 7.36237339276501362948391011349, 7.74193531672825538760351673707, 8.429855211758968290783980204213, 8.846429802662856184058682086077
