L(s) = 1 | + (0.984 − 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (0.766 + 0.642i)6-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + (−0.642 + 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (0.984 − 0.173i)17-s + i·18-s + (0.173 + 0.984i)21-s + (−0.642 − 0.766i)22-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (0.766 + 0.642i)6-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + (−0.642 + 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (0.984 − 0.173i)17-s + i·18-s + (0.173 + 0.984i)21-s + (−0.642 − 0.766i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.879153970 + 1.008662852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.879153970 + 1.008662852i\) |
\(L(1)\) |
\(\approx\) |
\(2.414037610 + 0.3786278221i\) |
\(L(1)\) |
\(\approx\) |
\(2.414037610 + 0.3786278221i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (0.642 + 0.766i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.984 - 0.173i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (-0.173 + 0.984i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.342 - 0.939i)T \) |
| 47 | \( 1 + (-0.984 - 0.173i)T \) |
| 53 | \( 1 + (-0.342 - 0.939i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.642 + 0.766i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.984 - 0.173i)T \) |
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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
−30.04759621799614061311053460642, −29.42593134479340744786003827428, −27.84690100041170690141763832017, −26.32142437792865592805740261683, −25.42361963274336171469293428154, −24.48777008375268519433589421590, −23.643082208225774233695692793348, −22.80617883979488516357759142476, −21.22104826855390283312097516641, −20.46489371589819485328127718660, −19.573727517743702451464162094200, −18.00297421793338313781442131760, −17.02618246940486342704473168927, −15.26409712269098395902227332199, −14.617264325845588608774835279643, −13.52855083962099343562903492937, −12.60214676733683613205251927819, −11.55132064617326635038935783713, −9.977889030074271093572859079603, −7.79567702098012848536681443950, −7.563926930717665818275263300667, −5.85445492495858179680753700602, −4.45246959165343248867336126382, −2.94042072090956638991781672363, −1.594104466949408611198744010629,
2.05152025849337246901375094102, 3.29820349004414119136071395334, 4.67658831945540031462427970070, 5.59632450785665200352903515368, 7.500613852253286327677286821340, 8.808534235822728137106897750036, 10.33829763851982138443980818420, 11.32626887708824569369743513246, 12.57792152240244386504874461540, 14.2251706781773141033485918794, 14.42936380510185449462220462985, 15.80680587362641595827843629057, 16.631101893604957229739292396538, 18.62847392854079064125131107281, 19.71038851118914437629292092011, 20.95934562098701735885440110443, 21.40415168921274021018278397082, 22.35395582311919393287923980186, 23.83265549903880146998250446090, 24.627017220605518403699239514428, 25.721047102624776256215287912278, 26.908080328577975043679499447613, 27.98093875898855011001213465691, 29.11595051824053979153169099696, 30.35507232720587853240979978556