L(s) = 1 | + (−0.770 − 0.637i)2-s + (0.844 + 0.535i)3-s + (0.187 + 0.982i)4-s + (−0.637 − 0.770i)5-s + (−0.309 − 0.951i)6-s + (−0.998 + 0.0627i)7-s + (0.481 − 0.876i)8-s + (0.425 + 0.904i)9-s + i·10-s + (0.904 − 0.425i)11-s + (−0.368 + 0.929i)12-s + (−0.0627 + 0.998i)13-s + (0.809 + 0.587i)14-s + (−0.125 − 0.992i)15-s + (−0.929 + 0.368i)16-s + (−0.309 + 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.770 − 0.637i)2-s + (0.844 + 0.535i)3-s + (0.187 + 0.982i)4-s + (−0.637 − 0.770i)5-s + (−0.309 − 0.951i)6-s + (−0.998 + 0.0627i)7-s + (0.481 − 0.876i)8-s + (0.425 + 0.904i)9-s + i·10-s + (0.904 − 0.425i)11-s + (−0.368 + 0.929i)12-s + (−0.0627 + 0.998i)13-s + (0.809 + 0.587i)14-s + (−0.125 − 0.992i)15-s + (−0.929 + 0.368i)16-s + (−0.309 + 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.329 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.329 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3386097516 + 0.4770703374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3386097516 + 0.4770703374i\) |
\(L(1)\) |
\(\approx\) |
\(0.6830833874 + 0.02938547658i\) |
\(L(1)\) |
\(\approx\) |
\(0.6830833874 + 0.02938547658i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 + (-0.770 - 0.637i)T \) |
| 3 | \( 1 + (0.844 + 0.535i)T \) |
| 5 | \( 1 + (-0.637 - 0.770i)T \) |
| 7 | \( 1 + (-0.998 + 0.0627i)T \) |
| 11 | \( 1 + (0.904 - 0.425i)T \) |
| 13 | \( 1 + (-0.0627 + 0.998i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.929 - 0.368i)T \) |
| 23 | \( 1 + (-0.968 + 0.248i)T \) |
| 29 | \( 1 + (-0.998 - 0.0627i)T \) |
| 31 | \( 1 + (0.0627 + 0.998i)T \) |
| 37 | \( 1 + (0.535 + 0.844i)T \) |
| 41 | \( 1 + (-0.951 + 0.309i)T \) |
| 43 | \( 1 + (-0.728 - 0.684i)T \) |
| 47 | \( 1 + (-0.728 + 0.684i)T \) |
| 53 | \( 1 + (0.982 + 0.187i)T \) |
| 59 | \( 1 + (-0.368 - 0.929i)T \) |
| 61 | \( 1 + (0.982 - 0.187i)T \) |
| 67 | \( 1 + (-0.844 + 0.535i)T \) |
| 71 | \( 1 + (0.535 - 0.844i)T \) |
| 73 | \( 1 + (0.248 + 0.968i)T \) |
| 79 | \( 1 + (0.968 + 0.248i)T \) |
| 83 | \( 1 + (-0.248 + 0.968i)T \) |
| 89 | \( 1 + (0.368 - 0.929i)T \) |
| 97 | \( 1 + (-0.187 - 0.982i)T \) |
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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
−29.58299745084750903807611207594, −27.97357352909708939284190873504, −27.03065493475159015453138416015, −26.08922782986560451262099765560, −25.39809788570798274594090549775, −24.50171634514737620564862525561, −23.19789868420505056268709600317, −22.45229610649176718089992398676, −20.22285446293654241128357145530, −19.70007987822673762563768941866, −18.75382057833965767830456104710, −17.94238745604937326161267344529, −16.46905283237554516016294847026, −15.25506497316353594088602564045, −14.66017415923040538109408406815, −13.34050863550180149848160015173, −11.87855214986776750096296051588, −10.27850695726340860420327860107, −9.30509961720987533192956642230, −8.0300132722667403960556393728, −7.07693930195992402453846959090, −6.23812339067925019909955517890, −3.821638919395119310137923981608, −2.33047201969241254598849170143, −0.28828211384869158179308660289,
1.77440561795706056816504916675, 3.49265704014731090482368340880, 4.23930948986499851331881122347, 6.74600526551484452100583684441, 8.309859808415618492705394706406, 8.98590823117991876756020274364, 9.91236930343672874933706045473, 11.32326880009109004230954583580, 12.52388162425584673335116096193, 13.55659187171242431425402599908, 15.23924089417114336401963854616, 16.35356205207301152546134482082, 16.92203164285255824127610123947, 18.92610220140377650486167464727, 19.53561408331375318103269792585, 20.15169371946849401046565175986, 21.44913277956027102499867808372, 22.133234926741968331451494814816, 23.93046934198140183903082380722, 25.1786621684503038507610190490, 26.058031268754121770081289278776, 26.9096223605294391395746758788, 27.89915539686483926159693344777, 28.623754228472018940323369104842, 29.95112774235319238102850359901