L(s) = 1 | + (0.366 − 0.366i)2-s + (−0.866 − 0.5i)3-s + 0.732i·4-s + (−0.5 + 0.133i)6-s + (0.633 + 0.633i)8-s + (0.499 + 0.866i)9-s + (0.366 − 0.633i)12-s − i·13-s − 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (−0.366 − 0.366i)26-s − 0.999i·27-s + ⋯ |
L(s) = 1 | + (0.366 − 0.366i)2-s + (−0.866 − 0.5i)3-s + 0.732i·4-s + (−0.5 + 0.133i)6-s + (0.633 + 0.633i)8-s + (0.499 + 0.866i)9-s + (0.366 − 0.633i)12-s − i·13-s − 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (−0.366 − 0.366i)26-s − 0.999i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.084475138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.084475138\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.409876902396886075578402610684, −8.364215319625513677218455031018, −7.73204663749347350574215234803, −7.07821217227158131102089797499, −6.16787685037953061378513476204, −5.22513946429756488004515141268, −4.63429578908161776856387020092, −3.46677617296152623034344502172, −2.61322163691978724226645359645, −1.29856665633053463670349467816,
0.915586976546599552619892365333, 2.38730044178192699650682811334, 4.15565750339888851861985690707, 4.35922411813746182554232787230, 5.45284178150281076795511102209, 6.02092574292142631439586123884, 6.74931738114314809733842765572, 7.40134775538651311991797932163, 8.785602891370929991581599454193, 9.411785078926859909802694611019