L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.342 + 0.939i)3-s + (0.766 − 0.642i)4-s − i·6-s + (0.984 − 0.173i)7-s + (−0.5 + 0.866i)8-s + (−0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.342 + 0.939i)12-s + (0.766 − 0.642i)13-s + (−0.866 + 0.5i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.939 + 0.342i)18-s + (0.342 − 0.939i)19-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.342 + 0.939i)3-s + (0.766 − 0.642i)4-s − i·6-s + (0.984 − 0.173i)7-s + (−0.5 + 0.866i)8-s + (−0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.342 + 0.939i)12-s + (0.766 − 0.642i)13-s + (−0.866 + 0.5i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.939 + 0.342i)18-s + (0.342 − 0.939i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7341742005 + 0.1061136530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7341742005 + 0.1061136530i\) |
\(L(1)\) |
\(\approx\) |
\(0.7079088913 + 0.1511617112i\) |
\(L(1)\) |
\(\approx\) |
\(0.7079088913 + 0.1511617112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (0.984 - 0.173i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.342 - 0.939i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.984 + 0.173i)T \) |
| 59 | \( 1 + (0.984 + 0.173i)T \) |
| 61 | \( 1 + (0.642 + 0.766i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.984 + 0.173i)T \) |
| 83 | \( 1 + (-0.642 + 0.766i)T \) |
| 89 | \( 1 + (-0.984 - 0.173i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.462416942671377913102146377923, −26.17083029279790511829386895805, −25.28769798756379712866340435531, −24.47946626337686920111083400086, −23.615312193229608808539167003099, −22.34472099389303815412798039445, −21.17457383819596444772311714774, −20.232376698771394456548533283888, −19.28401622753642328464296379632, −18.344805491751151422539836858530, −17.64946367003780750204507177039, −16.95871417351690107007005732366, −15.650492958436297019913019256840, −14.340749681511559554103192832403, −13.08873943017253620833004023201, −11.85842335767032667779844272542, −11.476682986411664070630338441809, −10.20434400779253811317326218095, −8.79957310952759427179230912592, −7.96921318883155407597717136010, −6.97060679304233829922199528551, −5.88481323224742391195312564130, −4.06910944566146819185172417206, −2.11995458240231623536888494397, −1.44003285423419141366087450649,
0.960848802261530792490641096469, 2.91716736668036147019791984446, 4.58647778750504902023255348707, 5.68360033929736579209624103604, 6.82635311134441417151694069330, 8.403390033352109993823491176432, 8.91402492219535355503475209614, 10.308680195364141443545281429551, 11.0548745042781213548155466381, 11.75490351555692061401639493748, 13.86024075441835493197565998006, 14.79459231951228589878524693058, 15.83652355657665594749489560863, 16.45932179786169968832251510709, 17.69740198854951780352587876533, 18.06380515117943460982484296807, 19.67206916930591311432200307045, 20.45340336635528262076798042233, 21.32489880196805398381340007526, 22.441577460759229402644123004086, 23.66409536741845191259206785001, 24.4829683704038516164888159883, 25.54130323810650106403554675768, 26.719034444042492763850204724467, 27.0942569713600301735431169453