L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 18-s + (−0.866 − 0.5i)19-s − i·21-s + (0.866 + 0.5i)22-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 18-s + (−0.866 − 0.5i)19-s − i·21-s + (0.866 + 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3440239996 + 0.09059023470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3440239996 + 0.09059023470i\) |
\(L(1)\) |
\(\approx\) |
\(0.6953233973 - 0.5284099893i\) |
\(L(1)\) |
\(\approx\) |
\(0.6953233973 - 0.5284099893i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 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
−19.39631011362636565796751453593, −18.9762518238169172542641251730, −18.2381842747892701416465116999, −17.599492886007639436819051435009, −16.51208956247995913754501630024, −16.02990825780905347390831338556, −15.223685570965941499684457942466, −14.87107055858877526522598149076, −14.09187861145893366722676614694, −13.336406504237080555381475139241, −12.57963678048433192327595853122, −11.24139328735014219364236738271, −10.51173016040862354773804765198, −9.92610328319718046943166406831, −8.834092077775252077453262124793, −8.498407528560494796806971874610, −7.99457183877317973810452050839, −6.968674267502631435674042572626, −6.00467728589785380630186048045, −5.222934690806571207999880467957, −4.51986280899641920488363625209, −3.53793529749428205003071215720, −2.28254022764623420457173205920, −1.746382456033506430075723156148, −0.07341061621598118389809384062,
0.80556206028019645326556119243, 1.983927417708072351708823879036, 2.305676040725431798862365012680, 3.443914086568141435946814333132, 4.20788984873910814932831498859, 4.908226329179404412186803339634, 6.48441610295597822105491408121, 7.42356293758535420315758715186, 7.81027078982070465317618709050, 8.5979093092747672018752002646, 9.40847367240124782502642768190, 10.11341906988589178791049466501, 10.869183827562024296957135521039, 11.62112304632410466332096398371, 12.52911569650530346641786687975, 13.29366277792051420378495452400, 13.5956008823810883048165228951, 14.50521356862344960106911816529, 15.39830308785140406637326276663, 16.22624845871530519799799634764, 17.40195730823077872374324941835, 17.72865999369219253164569217902, 18.418941118390652587932339312406, 19.234498293911575864518397324993, 19.8929614080920494559953344202