L(s) = 1 | − 2-s − 2.71·3-s + 4-s − 3.39·5-s + 2.71·6-s − 1.76·7-s − 8-s + 4.35·9-s + 3.39·10-s − 0.0397·11-s − 2.71·12-s + 5.20·13-s + 1.76·14-s + 9.19·15-s + 16-s + 2.00·17-s − 4.35·18-s − 5.14·19-s − 3.39·20-s + 4.79·21-s + 0.0397·22-s − 7.87·23-s + 2.71·24-s + 6.50·25-s − 5.20·26-s − 3.66·27-s − 1.76·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.56·3-s + 0.5·4-s − 1.51·5-s + 1.10·6-s − 0.668·7-s − 0.353·8-s + 1.45·9-s + 1.07·10-s − 0.0119·11-s − 0.782·12-s + 1.44·13-s + 0.472·14-s + 2.37·15-s + 0.250·16-s + 0.485·17-s − 1.02·18-s − 1.17·19-s − 0.758·20-s + 1.04·21-s + 0.00846·22-s − 1.64·23-s + 0.553·24-s + 1.30·25-s − 1.01·26-s − 0.706·27-s − 0.334·28-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
\(L(1)\) |
\(\approx\) |
\(0.1991477355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1991477355\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 + 0.0397T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 23 | \( 1 + 7.87T + 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.97T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 7.62T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 7.86T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 9.64T + 97T^{2} \) |
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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
−8.255824323446917431596625711747, −7.81739943078801165445606910326, −6.87982136969434965734480426597, −6.20628909473648413372956847725, −5.89009031072608094010236602831, −4.54510856608705633256004456070, −4.01132498396734534967498979493, −3.07897011621671799703618357823, −1.42150317154168501301665568958, −0.32792716818038109411820491578,
0.32792716818038109411820491578, 1.42150317154168501301665568958, 3.07897011621671799703618357823, 4.01132498396734534967498979493, 4.54510856608705633256004456070, 5.89009031072608094010236602831, 6.20628909473648413372956847725, 6.87982136969434965734480426597, 7.81739943078801165445606910326, 8.255824323446917431596625711747