L(s) = 1 | + (0.962 − 0.269i)2-s + (−0.974 − 0.225i)3-s + (0.854 − 0.519i)4-s + (−0.949 − 0.313i)5-s + (−0.998 + 0.0455i)6-s + (0.746 + 0.665i)7-s + (0.682 − 0.730i)8-s + (0.898 + 0.439i)9-s + (−0.998 − 0.0455i)10-s + (0.613 − 0.789i)11-s + (−0.949 + 0.313i)12-s + (−0.0682 + 0.997i)13-s + (0.898 + 0.439i)14-s + (0.854 + 0.519i)15-s + (0.460 − 0.887i)16-s + (−0.158 − 0.987i)17-s + ⋯ |
L(s) = 1 | + (0.962 − 0.269i)2-s + (−0.974 − 0.225i)3-s + (0.854 − 0.519i)4-s + (−0.949 − 0.313i)5-s + (−0.998 + 0.0455i)6-s + (0.746 + 0.665i)7-s + (0.682 − 0.730i)8-s + (0.898 + 0.439i)9-s + (−0.998 − 0.0455i)10-s + (0.613 − 0.789i)11-s + (−0.949 + 0.313i)12-s + (−0.0682 + 0.997i)13-s + (0.898 + 0.439i)14-s + (0.854 + 0.519i)15-s + (0.460 − 0.887i)16-s + (−0.158 − 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.282179914 - 0.8974554763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.282179914 - 0.8974554763i\) |
\(L(1)\) |
\(\approx\) |
\(1.271176605 - 0.4707030105i\) |
\(L(1)\) |
\(\approx\) |
\(1.271176605 - 0.4707030105i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 277 | \( 1 \) |
good | 2 | \( 1 + (0.962 - 0.269i)T \) |
| 3 | \( 1 + (-0.974 - 0.225i)T \) |
| 5 | \( 1 + (-0.949 - 0.313i)T \) |
| 7 | \( 1 + (0.746 + 0.665i)T \) |
| 11 | \( 1 + (0.613 - 0.789i)T \) |
| 13 | \( 1 + (-0.0682 + 0.997i)T \) |
| 17 | \( 1 + (-0.158 - 0.987i)T \) |
| 19 | \( 1 + (0.203 - 0.979i)T \) |
| 23 | \( 1 + (-0.949 - 0.313i)T \) |
| 29 | \( 1 + (0.0227 - 0.999i)T \) |
| 31 | \( 1 + (0.746 - 0.665i)T \) |
| 37 | \( 1 + (0.854 - 0.519i)T \) |
| 41 | \( 1 + (-0.775 + 0.631i)T \) |
| 43 | \( 1 + (-0.247 + 0.968i)T \) |
| 47 | \( 1 + (0.934 + 0.356i)T \) |
| 53 | \( 1 + (0.803 - 0.595i)T \) |
| 59 | \( 1 + (-0.990 + 0.136i)T \) |
| 61 | \( 1 + (-0.990 + 0.136i)T \) |
| 67 | \( 1 + (0.538 + 0.842i)T \) |
| 71 | \( 1 + (0.291 + 0.956i)T \) |
| 73 | \( 1 + (0.460 + 0.887i)T \) |
| 79 | \( 1 + (0.746 - 0.665i)T \) |
| 83 | \( 1 + (-0.998 + 0.0455i)T \) |
| 89 | \( 1 + (0.983 - 0.181i)T \) |
| 97 | \( 1 + (-0.974 - 0.225i)T \) |
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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
−25.71013471276686753978961054893, −24.53534959254064129689161611405, −23.70353520685356829242085899745, −23.16865993946951180869603538570, −22.42976028153868066863192868656, −21.64116253083206995865776366724, −20.40386735790072205797538452384, −19.88197075708885801674419028058, −18.25721910476610955903261888515, −17.28470276560355918159205232763, −16.58920346760501577115241533035, −15.40951301704427865348831007424, −14.96697465366687666540423929508, −13.8256552191683178842307269465, −12.2953954398127763761834219442, −12.11277741178417138897054116775, −10.843683316240690562706709988418, −10.33900106305950671634947255585, −8.13952011292732681506373611591, −7.34576338344766206361441152950, −6.3982104605417584377704645166, −5.21065440686934772174165304464, −4.24637751103907008965913285107, −3.57814772722566495675207434777, −1.54386942806241507061637198021,
1.00457855156891935040072735521, 2.46947507562608067286696380147, 4.16849381813400552439532456492, 4.74978710055647937573038086505, 5.883854662523796609950730297768, 6.84603041349185419699992223154, 7.97141813112546141260230117165, 9.4334349523139500916357705552, 11.053378391519661349226689682653, 11.732982611054462077358154740401, 11.8709059956709733032876568473, 13.23725049510341392272364330077, 14.2107341388892587381559061434, 15.39553060416873993406147088434, 16.09976235051715404038600832982, 16.93473465636655466941237038667, 18.36845898138415541937630948876, 19.12192381890004584051529619871, 20.10016778678787754712982887509, 21.2894601125537903230596527613, 21.91754511992634136857651773297, 22.78303330508513185106371697360, 23.67072923678452765632413800703, 24.44040399637302633925300841290, 24.68062588925539412304394304877