L(s) = 1 | + (0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s − 12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.913 − 0.406i)17-s + (−0.309 − 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)24-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s − 12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.913 − 0.406i)17-s + (−0.309 − 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2717321809 - 1.123953993i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2717321809 - 1.123953993i\) |
\(L(1)\) |
\(\approx\) |
\(0.7575883745 - 0.8120632944i\) |
\(L(1)\) |
\(\approx\) |
\(0.7575883745 - 0.8120632944i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 13 | \( 1 + (-0.913 + 0.406i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.978 + 0.207i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.669 - 0.743i)T \) |
| 53 | \( 1 + (-0.913 + 0.406i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.913 + 0.406i)T \) |
| 73 | \( 1 + (-0.669 + 0.743i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.104 - 0.994i)T \) |
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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
−22.08148207732845342139739254564, −21.457919582039328073561422581934, −20.427724261432704469729655644716, −19.44035882176086305203988316113, −18.94179492692482371100660739610, −18.04273235115583804007970647652, −17.26121399643991177861314641887, −16.26428497274755888376860322674, −15.49835712491208578236912373995, −15.0319727063882072593159484167, −14.4271746487289641956961914080, −13.320065917578135710516938199373, −12.91250730078414012258642098226, −11.94781143278071032819804712755, −10.51163971310752450129667164887, −9.536700556358400676370045560477, −9.02856138691752053813854070721, −8.29338525538519105731661108858, −7.40982083271334897592052772868, −6.67878270650223667898984663946, −5.505568519303662892288718178125, −4.84363633646451677624497901531, −3.71332328752044193792640089707, −2.92177571270713373873895292041, −1.78025366310093151121810821898,
0.38669548105053956392852348983, 1.74380477624891763722618500474, 2.486786445360931060470304601232, 3.47124899720782939069702596258, 4.208328200833034928911659921912, 5.00046223839079561612695240084, 6.57667821570935579176270370640, 7.37983366897937588827009930058, 8.29854238922147408152883550850, 9.32683982505674570310827241375, 9.692701572902388600647106318939, 10.70455230123629389943178254260, 11.45992909492251108931538693749, 12.71181788449689955874293443161, 12.98881099444818881754272786850, 13.9659280917952005329941509139, 14.43555604630141333912954899238, 15.31390599812259190663921006365, 16.53970424401363930624683433709, 17.35020076675370796683422640662, 18.312286802209790461273168327348, 18.99890905142055970661974260307, 19.728861247294421047259295417627, 20.24536011507259389777566145558, 20.81545888450599445416638862848