L(s) = 1 | + (0.187 + 0.982i)2-s + (0.425 + 0.904i)3-s + (−0.929 + 0.368i)4-s + (−0.187 + 0.982i)5-s + (−0.809 + 0.587i)6-s + (0.992 − 0.125i)7-s + (−0.535 − 0.844i)8-s + (−0.637 + 0.770i)9-s − 10-s + (0.637 − 0.770i)11-s + (−0.728 − 0.684i)12-s + (−0.992 − 0.125i)13-s + (0.309 + 0.951i)14-s + (−0.968 + 0.248i)15-s + (0.728 − 0.684i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
L(s) = 1 | + (0.187 + 0.982i)2-s + (0.425 + 0.904i)3-s + (−0.929 + 0.368i)4-s + (−0.187 + 0.982i)5-s + (−0.809 + 0.587i)6-s + (0.992 − 0.125i)7-s + (−0.535 − 0.844i)8-s + (−0.637 + 0.770i)9-s − 10-s + (0.637 − 0.770i)11-s + (−0.728 − 0.684i)12-s + (−0.992 − 0.125i)13-s + (0.309 + 0.951i)14-s + (−0.968 + 0.248i)15-s + (0.728 − 0.684i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2849386991 + 1.118071127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2849386991 + 1.118071127i\) |
\(L(1)\) |
\(\approx\) |
\(0.7184798533 + 0.9274126163i\) |
\(L(1)\) |
\(\approx\) |
\(0.7184798533 + 0.9274126163i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 + (0.187 + 0.982i)T \) |
| 3 | \( 1 + (0.425 + 0.904i)T \) |
| 5 | \( 1 + (-0.187 + 0.982i)T \) |
| 7 | \( 1 + (0.992 - 0.125i)T \) |
| 11 | \( 1 + (0.637 - 0.770i)T \) |
| 13 | \( 1 + (-0.992 - 0.125i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.728 + 0.684i)T \) |
| 23 | \( 1 + (0.876 - 0.481i)T \) |
| 29 | \( 1 + (0.992 + 0.125i)T \) |
| 31 | \( 1 + (-0.992 + 0.125i)T \) |
| 37 | \( 1 + (-0.425 + 0.904i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.0627 + 0.998i)T \) |
| 47 | \( 1 + (0.0627 - 0.998i)T \) |
| 53 | \( 1 + (0.929 + 0.368i)T \) |
| 59 | \( 1 + (-0.728 + 0.684i)T \) |
| 61 | \( 1 + (0.929 - 0.368i)T \) |
| 67 | \( 1 + (0.425 - 0.904i)T \) |
| 71 | \( 1 + (-0.425 - 0.904i)T \) |
| 73 | \( 1 + (-0.876 + 0.481i)T \) |
| 79 | \( 1 + (0.876 + 0.481i)T \) |
| 83 | \( 1 + (-0.876 - 0.481i)T \) |
| 89 | \( 1 + (-0.728 - 0.684i)T \) |
| 97 | \( 1 + (-0.929 + 0.368i)T \) |
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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
−29.47529700609699970018229052856, −28.61971685242908710870111099567, −27.68052560605568025667754421538, −26.63872973167742997267322511696, −24.967731066880605310830152318299, −24.214149819011592400201628758142, −23.38868858933810130202168692853, −21.96578610288045637388874727545, −20.79574066027766589950813390113, −19.94295465539455510995057048172, −19.34922202636197910782456805046, −17.76004675580060346482503890592, −17.37249325172139578496658874569, −15.09641718304878648261121064356, −14.14063888039878118524157797373, −12.9659586459725329063304492139, −12.17158990868276905999736618130, −11.31455195231088910188914803036, −9.413014215590783942194157584996, −8.634396208866087940727781980532, −7.32832777296406674262613888002, −5.30299286344406646219851926319, −4.19852644415880012463418989771, −2.32170489666040503982011461160, −1.25246122002676641808098371389,
2.95350357458542581999238843450, 4.240501493374071455986226746659, 5.37203434669652996033592785146, 6.96422890880826976955640075491, 8.07390653577983595451433992569, 9.19998989044767482860115322832, 10.52751777258695654146223954971, 11.73427211333315343832790235950, 13.82000950093026218482954581578, 14.454177309581895999228970131108, 15.17066296522452360391472173891, 16.33417192157023619886995374450, 17.39169226648007776988599885724, 18.53351761506886292610078127982, 19.82832951785892858642956125978, 21.30870219320560426699217821225, 22.12320005176369344414841210919, 22.8900433536140550074803232514, 24.36937001128742110317768739691, 25.109679753346131328695541065465, 26.47934634386566131916590694023, 27.00768937374226259650113480934, 27.51427345499947627433844769251, 29.460735894477357568042327180625, 30.96069001377246754753753838959