L(s) = 1 | − 2-s + 1.74·3-s + 4-s + 2.50·5-s − 1.74·6-s − 1.18·7-s − 8-s + 0.0572·9-s − 2.50·10-s − 1.65·11-s + 1.74·12-s + 1.18·14-s + 4.38·15-s + 16-s − 5.69·17-s − 0.0572·18-s + 19-s + 2.50·20-s − 2.07·21-s + 1.65·22-s − 1.77·23-s − 1.74·24-s + 1.28·25-s − 5.14·27-s − 1.18·28-s − 2.45·29-s − 4.38·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.00·3-s + 0.5·4-s + 1.12·5-s − 0.713·6-s − 0.448·7-s − 0.353·8-s + 0.0190·9-s − 0.792·10-s − 0.499·11-s + 0.504·12-s + 0.316·14-s + 1.13·15-s + 0.250·16-s − 1.38·17-s − 0.0134·18-s + 0.229·19-s + 0.560·20-s − 0.452·21-s + 0.353·22-s − 0.370·23-s − 0.356·24-s + 0.257·25-s − 0.990·27-s − 0.224·28-s − 0.456·29-s − 0.800·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 - 3.34T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 9.86T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 0.571T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.11T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 - 0.161T + 97T^{2} \) |
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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
−7.79856924146591060880525569398, −7.15191536579769251194723534999, −6.21319573707348282064191436705, −5.87490736598473470663642346543, −4.80275718551698313818762016345, −3.78435882749147773523510449517, −2.75529549254233850840584992600, −2.40131959779618747696096449891, −1.53596162877104204813177364949, 0,
1.53596162877104204813177364949, 2.40131959779618747696096449891, 2.75529549254233850840584992600, 3.78435882749147773523510449517, 4.80275718551698313818762016345, 5.87490736598473470663642346543, 6.21319573707348282064191436705, 7.15191536579769251194723534999, 7.79856924146591060880525569398