| L(s) = 1 | − 2-s − 0.732·3-s + 4-s − 5-s + 0.732·6-s − 3.46·7-s − 8-s − 2.46·9-s + 10-s + 2.73·11-s − 0.732·12-s + 1.26·13-s + 3.46·14-s + 0.732·15-s + 16-s + 4·17-s + 2.46·18-s + 5.46·19-s − 20-s + 2.53·21-s − 2.73·22-s + 0.535·23-s + 0.732·24-s + 25-s − 1.26·26-s + 4·27-s − 3.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.447·5-s + 0.298·6-s − 1.30·7-s − 0.353·8-s − 0.821·9-s + 0.316·10-s + 0.823·11-s − 0.211·12-s + 0.351·13-s + 0.925·14-s + 0.189·15-s + 0.250·16-s + 0.970·17-s + 0.580·18-s + 1.25·19-s − 0.223·20-s + 0.553·21-s − 0.582·22-s + 0.111·23-s + 0.149·24-s + 0.200·25-s − 0.248·26-s + 0.769·27-s − 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7852498166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7852498166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 0.535T + 23T^{2} \) |
| 29 | \( 1 + 0.732T + 29T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 - 5.46T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 4.39T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.63978072456699469926342236015, −7.03729571347948375054110715030, −6.34473625699751217322454304136, −5.86656172139827968094479834818, −5.14979639458039428738289261333, −3.97436563544769311786292170850, −3.26132148546509833139868371362, −2.81058823010722587021793110694, −1.35887787937066465149735544233, −0.52178561654437865727351164137,
0.52178561654437865727351164137, 1.35887787937066465149735544233, 2.81058823010722587021793110694, 3.26132148546509833139868371362, 3.97436563544769311786292170850, 5.14979639458039428738289261333, 5.86656172139827968094479834818, 6.34473625699751217322454304136, 7.03729571347948375054110715030, 7.63978072456699469926342236015
