| L(s) = 1 | − 0.732·3-s − 1.26i·7-s − 2.46·9-s + 3.46i·11-s + 3.46·13-s − 3.46i·17-s − 2i·19-s + 0.928i·21-s + 8.19i·23-s + 4·27-s − 9.46·31-s − 2.53i·33-s + 6·37-s − 2.53·39-s + 2.53·41-s + ⋯ |
| L(s) = 1 | − 0.422·3-s − 0.479i·7-s − 0.821·9-s + 1.04i·11-s + 0.960·13-s − 0.840i·17-s − 0.458i·19-s + 0.202i·21-s + 1.70i·23-s + 0.769·27-s − 1.69·31-s − 0.441i·33-s + 0.986·37-s − 0.406·39-s + 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245378060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245378060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 + 1.26iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 8.19iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.19iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.3iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 6.39iT - 97T^{2} \) |
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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
−9.271699234839596304784165759783, −9.015847147436326252942076484211, −7.55687965073468735720905853312, −7.34847334894503915981212498640, −6.11056806771401984182229514169, −5.51193255205727252338744014877, −4.52205395001125350466114157894, −3.59918204684581914792743069194, −2.44557022853871952542399460665, −1.03133021652598914611604034506,
0.63154940203184196003722781810, 2.21168102323827038235937310569, 3.31909520095100338431866862732, 4.21221432115153884411414181234, 5.59332107084300178945845068308, 5.86448894286268259036193452799, 6.65948532230546944241151172917, 7.930103581680832938509215246999, 8.645430558527555678479659101735, 9.003036863942755256568299135340
