L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.891 − 0.453i)3-s + (−0.951 − 0.309i)4-s + (−0.587 + 0.809i)6-s + (0.649 + 0.760i)7-s + (−0.453 + 0.891i)8-s + (0.587 + 0.809i)9-s + (0.852 + 0.522i)11-s + (0.707 + 0.707i)12-s + (0.923 + 0.382i)13-s + (0.852 − 0.522i)14-s + (0.809 + 0.587i)16-s + (0.972 − 0.233i)17-s + (0.891 − 0.453i)18-s + (−0.649 − 0.760i)19-s + ⋯ |
L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.891 − 0.453i)3-s + (−0.951 − 0.309i)4-s + (−0.587 + 0.809i)6-s + (0.649 + 0.760i)7-s + (−0.453 + 0.891i)8-s + (0.587 + 0.809i)9-s + (0.852 + 0.522i)11-s + (0.707 + 0.707i)12-s + (0.923 + 0.382i)13-s + (0.852 − 0.522i)14-s + (0.809 + 0.587i)16-s + (0.972 − 0.233i)17-s + (0.891 − 0.453i)18-s + (−0.649 − 0.760i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6394198915 - 1.429807543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6394198915 - 1.429807543i\) |
\(L(1)\) |
\(\approx\) |
\(0.8001671102 - 0.5739286182i\) |
\(L(1)\) |
\(\approx\) |
\(0.8001671102 - 0.5739286182i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.156 - 0.987i)T \) |
| 3 | \( 1 + (-0.891 - 0.453i)T \) |
| 7 | \( 1 + (0.649 + 0.760i)T \) |
| 11 | \( 1 + (0.852 + 0.522i)T \) |
| 13 | \( 1 + (0.923 + 0.382i)T \) |
| 17 | \( 1 + (0.972 - 0.233i)T \) |
| 19 | \( 1 + (-0.649 - 0.760i)T \) |
| 23 | \( 1 + (-0.0784 - 0.996i)T \) |
| 29 | \( 1 + (0.987 - 0.156i)T \) |
| 31 | \( 1 + (-0.233 - 0.972i)T \) |
| 37 | \( 1 + (-0.649 - 0.760i)T \) |
| 41 | \( 1 + (0.891 - 0.453i)T \) |
| 43 | \( 1 + (0.760 + 0.649i)T \) |
| 47 | \( 1 + (-0.156 - 0.987i)T \) |
| 53 | \( 1 + (0.987 - 0.156i)T \) |
| 59 | \( 1 + (0.453 - 0.891i)T \) |
| 61 | \( 1 + (-0.453 - 0.891i)T \) |
| 67 | \( 1 + (-0.156 + 0.987i)T \) |
| 71 | \( 1 + (-0.852 + 0.522i)T \) |
| 73 | \( 1 + (0.923 - 0.382i)T \) |
| 79 | \( 1 + (-0.987 - 0.156i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.233 + 0.972i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−17.66222371560823380068018690698, −17.230309049908656384249198635372, −16.516857437631222524516173402395, −16.23618319770438827127758621955, −15.39501764252959458956143217964, −14.767738140577892394854621723276, −14.08452444137199032426588380143, −13.63619909172330051490344713301, −12.65287717399203779681688484646, −12.08961342120026872359578932984, −11.353418095805818324825438587575, −10.51027599956674187172687249077, −10.1545944489800248873720109577, −9.15099225137274079114469153425, −8.52928070365122578679753846899, −7.79918531009126309822365837898, −7.09097653182886729493229879625, −6.30623073952288249168723349380, −5.836867237195402324760711152750, −5.18781901983462910224842390853, −4.34688564005031492958858317132, −3.8091049670202373468319093634, −3.27190066167371620532172153907, −1.22334014620385559148934359393, −1.09300911659611522842914015070,
0.5448225553078735743074924422, 1.33543815705097856956431543523, 2.04033524608404829665158080905, 2.64821148651330246103809670860, 3.910613497805014229111290585491, 4.4211389572052377299352915310, 5.145461828263942719001631300004, 5.86490032191327256383356479426, 6.43392511610043732505768567676, 7.35943132995912061863640264603, 8.317688016207570364163842641918, 8.83603399571814434083310764690, 9.62387846931369897220645091631, 10.40312727690847824176285302988, 11.10643245492385598281152510662, 11.51960068006919205212432080698, 12.19501910033874692557680529391, 12.555868727963188064211811072691, 13.32293929473969234749053948221, 14.14894342655013608107046303107, 14.59467207984775356396685327267, 15.46618447662248362712632479895, 16.307076019823533035975489575727, 17.06405555000796189623304847226, 17.7324741369635644079052292924