L(s) = 1 | + (0.866 − 0.5i)5-s − 7-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + 23-s + (0.5 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + 41-s + i·43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)5-s − 7-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + 23-s + (0.5 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + 41-s + i·43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3084818385 - 1.084181374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3084818385 - 1.084181374i\) |
\(L(1)\) |
\(\approx\) |
\(0.9791038506 - 0.2020077878i\) |
\(L(1)\) |
\(\approx\) |
\(0.9791038506 - 0.2020077878i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−20.17619780285744557029619528713, −19.41358679276978129638729742859, −18.67876617149202639795427408954, −18.15292702050600736416218417445, −17.29033463070196548293683585592, −16.591144405804651944996714318572, −15.793466643462070716611092143623, −15.177417062985178350740827542542, −14.11691015194721613236695854421, −13.63392898944829407020380357389, −12.85758258315007163392479744176, −12.2817113761926559024047975425, −10.93753151380615411672985867124, −10.55620911296018100541365710152, −9.77605033375249349955589928349, −9.0344751467223564953891857902, −8.198918733090703155426735421935, −7.04693912017428952591111037134, −6.55378752000631930380092379499, −5.64728307456446479409064922328, −5.07168838864304864485586912936, −3.60131926852236708052875248059, −3.06035062743526768802560329250, −2.18144828849131377697466435589, −1.06832038299345327862588720138,
0.216367623908084538941482772903, 1.13402999743108499708675522152, 2.48798384327724028839614996635, 2.83697963237924589060911696944, 4.19604210493905305086055527042, 5.09256876368897706540425240444, 5.675494099208819192968495844070, 6.68573998200541512719692644538, 7.28271256285267998313399926919, 8.37060355594704325877494969829, 9.41468511062684115644584634493, 9.58139208701577214988207994773, 10.50167970220899522943719077196, 11.40179524728821696186052060176, 12.40625112669388761108153930139, 13.100833322315549919286825646115, 13.44316536481454689823659857986, 14.34218530567467902451847564931, 15.37455026490870667435239101480, 16.07077383845560298558369835234, 16.51759804503509461611529537279, 17.74739326270635038943936181266, 17.8253820324494525347844691974, 18.927743906211334046880696741941, 19.63793921104051292827762675821