L(s) = 1 | + (0.998 − 0.0627i)2-s + (−0.368 + 0.929i)3-s + (0.992 − 0.125i)4-s + (0.0627 − 0.998i)5-s + (−0.309 + 0.951i)6-s + (0.844 − 0.535i)7-s + (0.982 − 0.187i)8-s + (−0.728 − 0.684i)9-s − i·10-s + (0.684 − 0.728i)11-s + (−0.248 + 0.968i)12-s + (−0.535 + 0.844i)13-s + (0.809 − 0.587i)14-s + (0.904 + 0.425i)15-s + (0.968 − 0.248i)16-s + (−0.309 − 0.951i)17-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0627i)2-s + (−0.368 + 0.929i)3-s + (0.992 − 0.125i)4-s + (0.0627 − 0.998i)5-s + (−0.309 + 0.951i)6-s + (0.844 − 0.535i)7-s + (0.982 − 0.187i)8-s + (−0.728 − 0.684i)9-s − i·10-s + (0.684 − 0.728i)11-s + (−0.248 + 0.968i)12-s + (−0.535 + 0.844i)13-s + (0.809 − 0.587i)14-s + (0.904 + 0.425i)15-s + (0.968 − 0.248i)16-s + (−0.309 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.158597648 - 0.3875325377i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.158597648 - 0.3875325377i\) |
\(L(1)\) |
\(\approx\) |
\(1.978488843 - 0.06092396747i\) |
\(L(1)\) |
\(\approx\) |
\(1.978488843 - 0.06092396747i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 + (0.998 - 0.0627i)T \) |
| 3 | \( 1 + (-0.368 + 0.929i)T \) |
| 5 | \( 1 + (0.0627 - 0.998i)T \) |
| 7 | \( 1 + (0.844 - 0.535i)T \) |
| 11 | \( 1 + (0.684 - 0.728i)T \) |
| 13 | \( 1 + (-0.535 + 0.844i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.968 + 0.248i)T \) |
| 23 | \( 1 + (0.637 + 0.770i)T \) |
| 29 | \( 1 + (0.844 + 0.535i)T \) |
| 31 | \( 1 + (0.535 + 0.844i)T \) |
| 37 | \( 1 + (-0.929 + 0.368i)T \) |
| 41 | \( 1 + (-0.951 - 0.309i)T \) |
| 43 | \( 1 + (-0.876 - 0.481i)T \) |
| 47 | \( 1 + (-0.876 + 0.481i)T \) |
| 53 | \( 1 + (0.125 - 0.992i)T \) |
| 59 | \( 1 + (-0.248 - 0.968i)T \) |
| 61 | \( 1 + (0.125 + 0.992i)T \) |
| 67 | \( 1 + (0.368 + 0.929i)T \) |
| 71 | \( 1 + (-0.929 - 0.368i)T \) |
| 73 | \( 1 + (-0.770 + 0.637i)T \) |
| 79 | \( 1 + (-0.637 + 0.770i)T \) |
| 83 | \( 1 + (0.770 + 0.637i)T \) |
| 89 | \( 1 + (0.248 - 0.968i)T \) |
| 97 | \( 1 + (-0.992 + 0.125i)T \) |
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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
−30.16939346868909182616207120927, −28.97521557082581231026815770145, −27.91367505990381372231410047050, −26.31568011759234397778210208863, −24.96346265959570954301898166061, −24.61947970639634266575037287144, −23.27956667569802627845311779479, −22.50783126907220563817419307200, −21.75106965779359464518987772791, −20.24969500371315255514304575899, −19.20578864129539061155424828379, −17.92426177651570934048674596005, −17.14716659797916846697466305135, −15.25886790234559461283036932911, −14.62672109206598652694124116592, −13.52391377915725682491992857825, −12.26348716945271561936957734898, −11.542932694252953137777143220417, −10.41380941368943194478886316635, −8.06114466469589645370426023193, −7.031047189412027588932965054612, −6.05385187632329316110398877826, −4.80912824766302432457595530706, −2.90019481439899593853720458684, −1.76524528537434114004961734014,
1.2633334583474031373924158176, 3.45983571501983651277737015196, 4.70834135780160551885288174707, 5.253918520975156874740639497356, 6.89593136129938864094398606181, 8.6591630213810792955702243938, 9.995655958871190445351104472553, 11.48639451980185246870025687271, 11.8860298793875887186260380284, 13.684283957808479383726708490301, 14.372112555312253783722374841964, 15.797076021723125329465565538817, 16.56023866441703365131334505116, 17.43135327195679494553876352647, 19.630420198180040330050135341831, 20.56023916991029679334506650895, 21.28192311746190006361761549034, 22.13854706706544690920011040301, 23.37336889891623325648452007163, 24.207191008379084900463198920447, 25.09600037187474181685790541132, 26.77078058902921371099534985345, 27.525702860708308020105884628755, 28.86406626403537472561159159637, 29.38649383684297752601180726270