L(s) = 1 | + 2.61·5-s + (1.25 + 2.32i)7-s + 4.29i·11-s − 1.53i·13-s + 0.448·17-s − 1.92i·19-s − 0.737i·23-s + 1.82·25-s − 1.41i·29-s + 6.58i·31-s + (3.29 + 6.08i)35-s + 2.82·37-s + 6.94·41-s + 9.64·43-s − 2.72·47-s + ⋯ |
L(s) = 1 | + 1.16·5-s + (0.475 + 0.879i)7-s + 1.29i·11-s − 0.424i·13-s + 0.108·17-s − 0.442i·19-s − 0.153i·23-s + 0.365·25-s − 0.262i·29-s + 1.18i·31-s + (0.556 + 1.02i)35-s + 0.464·37-s + 1.08·41-s + 1.47·43-s − 0.397·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.544718909\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.544718909\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.25 - 2.32i)T \) |
good | 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 - 4.29iT - 11T^{2} \) |
| 13 | \( 1 + 1.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.448T + 17T^{2} \) |
| 19 | \( 1 + 1.92iT - 19T^{2} \) |
| 23 | \( 1 + 0.737iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.58iT - 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 8.28iT - 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 11.0iT - 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.75iT - 97T^{2} \) |
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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
−8.724723964312867474718547214214, −7.82648016753161574417590508854, −7.09773595621007463661461515985, −6.24581227308613011653110829591, −5.57387275169404351696645887022, −4.98936274758509228224720065978, −4.16024853595578120664251303332, −2.73550614719050498252577410819, −2.23738271328370743289660397606, −1.28736443417760432745388530835,
0.75750078974457567776074739565, 1.72264022149180614917342023881, 2.69625515438988869488370844902, 3.75895155671489623227565081582, 4.47896595317292762723283002503, 5.59950628187789452736072342345, 5.93010987860384388757624740040, 6.77664415590219141075379171323, 7.64964776377807371185210637526, 8.265742822886785296095940288069