L(s) = 1 | + 0.206·5-s − i·7-s + 6.42i·11-s + 3.52i·13-s − 3.90i·17-s + 5.48·19-s + 0.880·23-s − 4.95·25-s − 3.20·29-s − 0.0631i·31-s − 0.206i·35-s + 9.91i·37-s + 5.94i·41-s + 1.13·43-s − 6.81·47-s + ⋯ |
L(s) = 1 | + 0.0925·5-s − 0.377i·7-s + 1.93i·11-s + 0.977i·13-s − 0.947i·17-s + 1.25·19-s + 0.183·23-s − 0.991·25-s − 0.595·29-s − 0.0113i·31-s − 0.0349i·35-s + 1.63i·37-s + 0.928i·41-s + 0.173·43-s − 0.993·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.127522634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.127522634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.206T + 5T^{2} \) |
| 11 | \( 1 - 6.42iT - 11T^{2} \) |
| 13 | \( 1 - 3.52iT - 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 - 0.880T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 0.0631iT - 31T^{2} \) |
| 37 | \( 1 - 9.91iT - 37T^{2} \) |
| 41 | \( 1 - 5.94iT - 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 + 6.81T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 4.23iT - 59T^{2} \) |
| 61 | \( 1 + 12.3iT - 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 8.27T + 73T^{2} \) |
| 79 | \( 1 - 4.86iT - 79T^{2} \) |
| 83 | \( 1 - 7.51iT - 83T^{2} \) |
| 89 | \( 1 + 4.44iT - 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.075658103732779170012576310513, −7.66193920928119539446429421003, −6.81893626911814583681808418766, −6.55410510142285908657621182151, −5.19861744235569355221674344917, −4.82776891414302879900491984075, −4.04527605331574747313578730783, −3.11628395296194595157635997501, −2.08291317055201699230234822153, −1.35532439509659959904436438744,
0.28676629360772184537567172587, 1.36520391432748398583958054498, 2.56193847049017460948852654683, 3.39505496243778003163404231663, 3.86740534306640103794536296494, 5.24027030404332222139815064524, 5.70173561035203690173728051176, 6.11474634091348412670637550339, 7.17604942846863216269319218146, 8.028971389533660371300506389451