L(s) = 1 | + 0.954·2-s − 1.64·3-s − 1.08·4-s + 5-s − 1.57·6-s − 7-s − 2.94·8-s − 0.293·9-s + 0.954·10-s + 4.62·11-s + 1.79·12-s − 2.12·13-s − 0.954·14-s − 1.64·15-s − 0.638·16-s − 2.62·17-s − 0.279·18-s − 4.08·19-s − 1.08·20-s + 1.64·21-s + 4.41·22-s + 4.30·23-s + 4.85·24-s + 25-s − 2.02·26-s + 5.41·27-s + 1.08·28-s + ⋯ |
L(s) = 1 | + 0.675·2-s − 0.949·3-s − 0.544·4-s + 0.447·5-s − 0.641·6-s − 0.377·7-s − 1.04·8-s − 0.0976·9-s + 0.301·10-s + 1.39·11-s + 0.516·12-s − 0.589·13-s − 0.255·14-s − 0.424·15-s − 0.159·16-s − 0.635·17-s − 0.0659·18-s − 0.938·19-s − 0.243·20-s + 0.359·21-s + 0.940·22-s + 0.897·23-s + 0.990·24-s + 0.200·25-s − 0.397·26-s + 1.04·27-s + 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 0.954T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 + 0.713T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 + 5.11T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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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
−7.12540646406033585296043992759, −6.46254197470671268923584380997, −6.07434229904080983722359077121, −5.44124460395020130383257433973, −4.60330237953193799870461556063, −4.23119090148812779443177732228, −3.23862772963691808681786409230, −2.38598433351784534399255006734, −1.05748142845701655287972355068, 0,
1.05748142845701655287972355068, 2.38598433351784534399255006734, 3.23862772963691808681786409230, 4.23119090148812779443177732228, 4.60330237953193799870461556063, 5.44124460395020130383257433973, 6.07434229904080983722359077121, 6.46254197470671268923584380997, 7.12540646406033585296043992759