L(s) = 1 | + (0.841 + 0.540i)2-s + 3-s + (0.415 + 0.909i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + 9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)14-s + (−0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (−0.959 + 0.281i)17-s + (0.841 + 0.540i)18-s + ⋯ |
L(s) = 1 | + (0.841 + 0.540i)2-s + 3-s + (0.415 + 0.909i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + 9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)14-s + (−0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (−0.959 + 0.281i)17-s + (0.841 + 0.540i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.963550484 + 0.4673647071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.963550484 + 0.4673647071i\) |
\(L(1)\) |
\(\approx\) |
\(1.851146532 + 0.3641541468i\) |
\(L(1)\) |
\(\approx\) |
\(1.851146532 + 0.3641541468i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (-0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.654 + 0.755i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + (0.415 - 0.909i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (-0.142 + 0.989i)T \) |
| 47 | \( 1 + (0.841 - 0.540i)T \) |
| 53 | \( 1 + (-0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (-0.959 - 0.281i)T \) |
| 73 | \( 1 + (-0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.654 - 0.755i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.142 + 0.989i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.35025659355156785223151474253, −28.06650296582281583489975441268, −26.94541795695253241072831709047, −25.69920040201296585520340932158, −25.0637690817309914701173370790, −23.834553195045223633096360383190, −22.52917169695938488196149826079, −22.0101815331457734940514778397, −20.8145722892727983543893527738, −19.83381299011550398837036577847, −18.95894457341483024978833061368, −18.2280050721085033402483899064, −15.80975875189256641565989398439, −15.23652496532244966405452044508, −14.31509664841571157836938251349, −13.24632614657597472774516872991, −12.34873935639678668598758053013, −10.861252404707179492183663719782, −9.965766793095050840163281608165, −8.62812810769823624101991990162, −7.00583453961443350429246221077, −5.95877851873458357656797204605, −4.127473907468224388251382589314, −3.02635340464008995703770642635, −2.23338409626685844939198256114,
2.0836239353917443368655041692, 3.853197606562159406171606997067, 4.36976080201992622155435320397, 6.21535283548911252260237227969, 7.4129403981781401615179505076, 8.50136464553820349781787498075, 9.51972141083152244344905198870, 11.35843012476731755940385017528, 12.97223656027831398984886378524, 13.23436963753698010441811793801, 14.38917051793100121189909283528, 15.625623234715500638619357555717, 16.332512619474735750412166577153, 17.40977754482196839589777180406, 19.279493249049610243717408880867, 20.12692138416666942720681391811, 20.96571167457529935478179890081, 21.90826381883849867059580896305, 23.405871919029645148988293196214, 24.05197186003123354359129180007, 24.99072115845674285343634308774, 26.034717799102593666816101594056, 26.57252242642424974329808318120, 28.12718088433311927402774577301, 29.481570471389556437300828840176