L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.5 + 0.866i)12-s + (0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (0.809 − 0.587i)27-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.5 + 0.866i)12-s + (0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (0.809 − 0.587i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7969632941 - 0.3039086892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7969632941 - 0.3039086892i\) |
\(L(1)\) |
\(\approx\) |
\(0.7101941555 - 0.3918875518i\) |
\(L(1)\) |
\(\approx\) |
\(0.7101941555 - 0.3918875518i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (0.104 + 0.994i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.913 - 0.406i)T \) |
| 37 | \( 1 + (-0.669 + 0.743i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.978 - 0.207i)T \) |
| 53 | \( 1 + (-0.913 + 0.406i)T \) |
| 59 | \( 1 + (-0.978 - 0.207i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.978 + 0.207i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−24.70761470992347083394592111149, −23.442708282123014869905522724697, −23.048552462163331064342182826605, −22.27286303146828373691981763004, −21.29192633403198344229327496988, −20.591023151013128236361352128300, −19.08086633729713784677323762808, −18.098236252853682158280314370428, −17.40444483040782863073617294085, −16.55041659124478070855760318749, −15.77992278588670740413717247605, −15.145715134365143795570396499298, −14.147909103799944107792328962239, −13.09852350067617216862003439507, −12.12467790357490722866996379541, −10.939375637721343271740936102466, −10.01879805713945168295962716506, −9.02292925339269055573087890415, −8.15298878063183782893733636393, −6.78637335726616960359187050683, −6.07594717945580718061044998647, −5.03856773661411502119734717008, −4.27004747564156954431225500009, −3.08994656667596580494770863740, −0.641070552006673849278949736650,
1.20595864230173805113394400267, 2.06412446333604644500550330562, 3.50014959440291844537428035080, 4.63714724523008816254005952533, 5.74392535716836395288955151759, 6.66064979122560116096572869856, 8.08121469273748552513887791775, 8.902870419472983127794852332207, 10.295281884630723911865046425478, 10.96566953128533720154736029603, 11.83564032010147290009394662931, 12.64807497596990106283359376559, 13.40878641394990241215539108156, 14.22415047423974613346105574434, 15.5094059641895415626508310566, 16.93952665803484413663773522449, 17.43062507508339843568405154551, 18.59087946472816341376157896311, 19.02056977094933926567189331522, 19.892865826896617475400550833978, 21.07475302300163317693411606136, 21.72306903764520596482503498372, 22.669275571183345668559682138261, 23.56176503091781830591471381550, 23.87840923010596320835541196497