L(s) = 1 | + 2-s + 0.853·3-s + 4-s + 2.63·5-s + 0.853·6-s − 0.512·7-s + 8-s − 2.27·9-s + 2.63·10-s − 5.31·11-s + 0.853·12-s − 6.23·13-s − 0.512·14-s + 2.25·15-s + 16-s − 2.45·17-s − 2.27·18-s − 4.33·19-s + 2.63·20-s − 0.437·21-s − 5.31·22-s + 4.35·23-s + 0.853·24-s + 1.96·25-s − 6.23·26-s − 4.50·27-s − 0.512·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.492·3-s + 0.5·4-s + 1.18·5-s + 0.348·6-s − 0.193·7-s + 0.353·8-s − 0.757·9-s + 0.834·10-s − 1.60·11-s + 0.246·12-s − 1.72·13-s − 0.137·14-s + 0.581·15-s + 0.250·16-s − 0.594·17-s − 0.535·18-s − 0.994·19-s + 0.590·20-s − 0.0955·21-s − 1.13·22-s + 0.907·23-s + 0.174·24-s + 0.393·25-s − 1.22·26-s − 0.866·27-s − 0.0969·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 0.853T + 3T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 + 0.512T + 7T^{2} \) |
| 11 | \( 1 + 5.31T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 2.45T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 - 4.35T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 - 2.64T + 43T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 1.98T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 6.21T + 97T^{2} \) |
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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
−7.84774216087125851348256806928, −7.43974534477517228954023671812, −6.39823827348151031816788659667, −5.73701878357726078679313513813, −5.13024610852911013587147300365, −4.45700297834270160655739602456, −3.10300760789028154916911779260, −2.47338332438014584912370757115, −2.08548206091404015317691650476, 0,
2.08548206091404015317691650476, 2.47338332438014584912370757115, 3.10300760789028154916911779260, 4.45700297834270160655739602456, 5.13024610852911013587147300365, 5.73701878357726078679313513813, 6.39823827348151031816788659667, 7.43974534477517228954023671812, 7.84774216087125851348256806928