L(s) = 1 | + (0.125 − 0.992i)2-s + (0.684 + 0.728i)3-s + (−0.968 − 0.248i)4-s + (−0.992 + 0.125i)5-s + (0.809 − 0.587i)6-s + (0.904 − 0.425i)7-s + (−0.368 + 0.929i)8-s + (−0.0627 + 0.998i)9-s + i·10-s + (0.998 + 0.0627i)11-s + (−0.481 − 0.876i)12-s + (0.425 − 0.904i)13-s + (−0.309 − 0.951i)14-s + (−0.770 − 0.637i)15-s + (0.876 + 0.481i)16-s + (0.809 + 0.587i)17-s + ⋯ |
L(s) = 1 | + (0.125 − 0.992i)2-s + (0.684 + 0.728i)3-s + (−0.968 − 0.248i)4-s + (−0.992 + 0.125i)5-s + (0.809 − 0.587i)6-s + (0.904 − 0.425i)7-s + (−0.368 + 0.929i)8-s + (−0.0627 + 0.998i)9-s + i·10-s + (0.998 + 0.0627i)11-s + (−0.481 − 0.876i)12-s + (0.425 − 0.904i)13-s + (−0.309 − 0.951i)14-s + (−0.770 − 0.637i)15-s + (0.876 + 0.481i)16-s + (0.809 + 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.990496601 - 0.5531527292i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.990496601 - 0.5531527292i\) |
\(L(1)\) |
\(\approx\) |
\(1.310919743 - 0.3340005020i\) |
\(L(1)\) |
\(\approx\) |
\(1.310919743 - 0.3340005020i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 + (0.125 - 0.992i)T \) |
| 3 | \( 1 + (0.684 + 0.728i)T \) |
| 5 | \( 1 + (-0.992 + 0.125i)T \) |
| 7 | \( 1 + (0.904 - 0.425i)T \) |
| 11 | \( 1 + (0.998 + 0.0627i)T \) |
| 13 | \( 1 + (0.425 - 0.904i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.876 - 0.481i)T \) |
| 23 | \( 1 + (0.187 + 0.982i)T \) |
| 29 | \( 1 + (0.904 + 0.425i)T \) |
| 31 | \( 1 + (-0.425 - 0.904i)T \) |
| 37 | \( 1 + (0.728 + 0.684i)T \) |
| 41 | \( 1 + (-0.587 - 0.809i)T \) |
| 43 | \( 1 + (-0.535 + 0.844i)T \) |
| 47 | \( 1 + (-0.535 - 0.844i)T \) |
| 53 | \( 1 + (-0.248 - 0.968i)T \) |
| 59 | \( 1 + (-0.481 + 0.876i)T \) |
| 61 | \( 1 + (-0.248 + 0.968i)T \) |
| 67 | \( 1 + (-0.684 + 0.728i)T \) |
| 71 | \( 1 + (0.728 - 0.684i)T \) |
| 73 | \( 1 + (0.982 - 0.187i)T \) |
| 79 | \( 1 + (-0.187 + 0.982i)T \) |
| 83 | \( 1 + (-0.982 - 0.187i)T \) |
| 89 | \( 1 + (0.481 + 0.876i)T \) |
| 97 | \( 1 + (0.968 + 0.248i)T \) |
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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
−30.37700004607146010048568020096, −28.48974020829642489706154044330, −27.221541779372304469861314914005, −26.68887627499297853240417489958, −25.21476205428597924808925729053, −24.65193811976607860226905129932, −23.73744819733892671211249245395, −22.9265513638171847905614924852, −21.41032102916061190696691090118, −20.14763213880289300108750808176, −18.85522720520187598970503560738, −18.270901033427980261225908174462, −16.84225705689129643552389958079, −15.727350612255040427373552878678, −14.46750671594210800104726031075, −14.12018327595377806787472833176, −12.41383011623794546792595881647, −11.67609317667166170741485231830, −9.24828271771426117892596702120, −8.38206792953602654813036295550, −7.512250388545416831665240877084, −6.40149528554953548986511253688, −4.68333254873790358399603895268, −3.42296276767922940093052203568, −1.13935846359545651884988348010,
1.23995839979314684101512346840, 3.19463335338833943405496593559, 3.990786743651080037962260235972, 5.14705613810112457686877640995, 7.727637691868041746069898478223, 8.60451316127172127762871447107, 9.93801795192573621555336213380, 11.029526138093325391007816269809, 11.837487681426458288638187332126, 13.41627654425433895259577841100, 14.521052983857279586200844746048, 15.2497830519619839449812697802, 16.829147264524068175286339323049, 18.1844237567815831937082533311, 19.55694629249541493545999829326, 20.04084530380393345193175579739, 20.987006707829324722710912222729, 22.06040730693709289179922684974, 23.04013309930807879062078282730, 24.16033027668624830297919699611, 25.69830777186295867706665740408, 27.03793989404435216438980650731, 27.417650466828579042178968456223, 28.15198892158290607939101482993, 29.99071270841209841714869499574