L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.978 − 0.207i)3-s + (−0.104 + 0.994i)4-s + (−0.809 − 0.587i)6-s + (0.913 − 0.406i)7-s + (0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.669 − 0.743i)11-s + (0.104 + 0.994i)12-s + (0.913 + 0.406i)13-s + (−0.913 − 0.406i)14-s + (−0.978 − 0.207i)16-s + 17-s + (−0.913 − 0.406i)18-s + (0.104 + 0.994i)19-s + ⋯ |
L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.978 − 0.207i)3-s + (−0.104 + 0.994i)4-s + (−0.809 − 0.587i)6-s + (0.913 − 0.406i)7-s + (0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.669 − 0.743i)11-s + (0.104 + 0.994i)12-s + (0.913 + 0.406i)13-s + (−0.913 − 0.406i)14-s + (−0.978 − 0.207i)16-s + 17-s + (−0.913 − 0.406i)18-s + (0.104 + 0.994i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.156913483 - 1.326642214i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.156913483 - 1.326642214i\) |
\(L(1)\) |
\(\approx\) |
\(1.232098311 - 0.5180889452i\) |
\(L(1)\) |
\(\approx\) |
\(1.232098311 - 0.5180889452i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 7 | \( 1 + (0.913 - 0.406i)T \) |
| 11 | \( 1 + (-0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.913 + 0.406i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.104 + 0.994i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + (-0.809 + 0.587i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.978 + 0.207i)T \) |
| 89 | \( 1 + (-0.669 - 0.743i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−17.79763374005711266069922882743, −17.345972013791922367844259855476, −16.354666844555493252395190054416, −15.61746133136436739201648031198, −15.40284030497335256825472742916, −14.58028989286762355910977830804, −14.3018229105422270552109329125, −13.26188364534255379811710357138, −12.96440548249899979963086606913, −11.69890737549793433226205918879, −11.00613457390059617168832507, −10.287712195444758033179486911762, −9.74119832834465867483019726072, −8.96198546559813183829479761887, −8.408466069979076080329201449393, −7.9265978368135986893152210883, −7.28924432668809526412479425666, −6.62308415689051237715583233011, −5.427649393535805670265892630227, −5.096867320209822215392593890133, −4.31406760644258871782212570354, −3.26492437134885273001324913504, −2.414912254936979814630843559951, −1.68872688404416575118980989776, −0.9127334816916904413640223383,
0.91407464647325577439319725089, 1.42897318046772220340675741813, 2.12741973979026907620045496971, 3.19708118541595668559308138916, 3.46233253440488637516690244372, 4.32506631817346747502139001832, 5.17694319074060831192903377422, 6.256147169688621319283663304166, 7.28811555637016777206176975109, 7.72022960647880826505435218167, 8.434132544910599679346223021119, 8.698626629963720502610425792978, 9.64083067640413176775467708165, 10.29556567501985363239048199227, 10.88666815517822774376022908417, 11.54988764651020933963049373014, 12.28028906283045252809594732162, 13.06803425897423125522026685380, 13.614995547493050374776908685181, 14.122386607318468620035560141491, 14.88292280923101735508863697965, 15.67757114125675889873763937132, 16.556730890745778573247827904806, 16.853370995017245346523607744630, 17.93140513543027911106937082003