| L(s) = 1 | + 1.93i·2-s − 1.73·4-s + (−1 + 2.44i)7-s + 0.517i·8-s − 5.27i·11-s − 6.69i·13-s + (−4.73 − 1.93i)14-s − 4.46·16-s + 3.46·17-s − 6.69i·19-s + 10.1·22-s − 1.41i·23-s + 12.9·26-s + (1.73 − 4.24i)28-s + 1.41i·29-s + ⋯ |
| L(s) = 1 | + 1.36i·2-s − 0.866·4-s + (−0.377 + 0.925i)7-s + 0.183i·8-s − 1.59i·11-s − 1.85i·13-s + (−1.26 − 0.516i)14-s − 1.11·16-s + 0.840·17-s − 1.53i·19-s + 2.17·22-s − 0.294i·23-s + 2.53·26-s + (0.327 − 0.801i)28-s + 0.262i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250898855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250898855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| good | 2 | \( 1 - 1.93iT - 2T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 + 6.69iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.69iT - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 1.79iT - 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 4.52iT - 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 3.58iT - 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 9.41iT - 71T^{2} \) |
| 73 | \( 1 - 6.69iT - 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 10.2iT - 97T^{2} \) |
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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
−9.038116866723450077465757443057, −8.453284453433401224447395250611, −7.941446137577236025345574789639, −6.97392523208875225227589659536, −6.10090280884624670090084308070, −5.57248216584163989013969634864, −5.01789473856834667069167758970, −3.35697725953070575328903991881, −2.66505240187836457200225993604, −0.49742486219028495809722545138,
1.40445548162736190775702066392, 2.04041108916658314969530758287, 3.42709329611219382762092757245, 4.08831125736766266272538288536, 4.77023597517994443658583942541, 6.28424321447028887858397513100, 7.06105069578505861167658350834, 7.70826221793098356871236776304, 9.066168918453785058282062064930, 9.744923452687198842391716571209
