| L(s) = 1 | − 0.732i·3-s − 1.26·7-s + 2.46·9-s + 3.46i·11-s + 3.46i·13-s − 3.46·17-s + 2i·19-s + 0.928i·21-s − 8.19·23-s − 4i·27-s − 9.46·31-s + 2.53·33-s − 6i·37-s + 2.53·39-s + 2.53·41-s + ⋯ |
| L(s) = 1 | − 0.422i·3-s − 0.479·7-s + 0.821·9-s + 1.04i·11-s + 0.960i·13-s − 0.840·17-s + 0.458i·19-s + 0.202i·21-s − 1.70·23-s − 0.769i·27-s − 1.69·31-s + 0.441·33-s − 0.986i·37-s + 0.406·39-s + 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9267251933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9267251933\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.732iT - 3T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4.73iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−9.597319049807815665446440370236, −9.016234936880634690360374570293, −7.82510917362072169206666260476, −7.23178649093466573525216516984, −6.54982041597820559567011241123, −5.72495468034720360008856466359, −4.36156406636242263863830466630, −3.99733779439817723181855403385, −2.36071409947482893284312822335, −1.60447023758483719775756512531,
0.34267446449125540644934033319, 2.01353434820075179885798404821, 3.32242106923152186483872695255, 3.95983321760789105428529623148, 5.05734636538964618529694229974, 5.89347622653997201109091335189, 6.71232668698640180797667246562, 7.62064198156454184464574209902, 8.455600646678025268710941267333, 9.236016914716632101969488478226
