L(s) = 1 | + 1.73·3-s + 3.46i·7-s + 2.99·9-s + 2i·13-s + 3.46·19-s + 5.99i·21-s − 5·25-s + 5.19·27-s + 10.3i·31-s − 10i·37-s + 3.46i·39-s + 10.3·43-s − 4.99·49-s + 5.99·57-s − 14i·61-s + ⋯ |
L(s) = 1 | + 1.00·3-s + 1.30i·7-s + 0.999·9-s + 0.554i·13-s + 0.794·19-s + 1.30i·21-s − 25-s + 1.00·27-s + 1.86i·31-s − 1.64i·37-s + 0.554i·39-s + 1.58·43-s − 0.714·49-s + 0.794·57-s − 1.79i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01692 + 0.835437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01692 + 0.835437i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.22654891479297754034386346272, −9.227003619870129554422183187084, −8.963434064742152129354526862216, −7.952961568246118900465773304452, −7.13540129664529171036333224101, −6.00884931725512877115327692953, −5.01717303883842896772106303660, −3.80607652190433463268422928109, −2.75816835518076104426723789851, −1.78081088725830406099711628756,
1.10152930121939473163336821913, 2.62523968266781872014428847440, 3.74309387477081638887245261975, 4.43247110074070987928508827188, 5.82195847638298504143501597175, 7.08100181402466285672416995097, 7.65627223821754878042602810443, 8.348205826487839323201098497833, 9.561787965451448481708972119251, 9.987776662992207907731092552525