L(s) = 1 | + (0.516 − 0.856i)2-s + (−0.809 − 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.921 + 0.389i)6-s + (−0.998 − 0.0570i)8-s + (0.309 + 0.951i)9-s + (−0.142 + 0.989i)12-s + (0.985 − 0.170i)13-s + (−0.564 + 0.825i)16-s + (−0.993 + 0.113i)17-s + (0.974 + 0.226i)18-s + (−0.870 + 0.491i)19-s + (0.959 + 0.281i)23-s + (0.774 + 0.633i)24-s + (0.362 − 0.931i)26-s + (0.309 − 0.951i)27-s + ⋯ |
L(s) = 1 | + (0.516 − 0.856i)2-s + (−0.809 − 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.921 + 0.389i)6-s + (−0.998 − 0.0570i)8-s + (0.309 + 0.951i)9-s + (−0.142 + 0.989i)12-s + (0.985 − 0.170i)13-s + (−0.564 + 0.825i)16-s + (−0.993 + 0.113i)17-s + (0.974 + 0.226i)18-s + (−0.870 + 0.491i)19-s + (0.959 + 0.281i)23-s + (0.774 + 0.633i)24-s + (0.362 − 0.931i)26-s + (0.309 − 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2700009087 - 1.196237937i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2700009087 - 1.196237937i\) |
\(L(1)\) |
\(\approx\) |
\(0.7182588299 - 0.6210224890i\) |
\(L(1)\) |
\(\approx\) |
\(0.7182588299 - 0.6210224890i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.516 - 0.856i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.985 - 0.170i)T \) |
| 17 | \( 1 + (-0.993 + 0.113i)T \) |
| 19 | \( 1 + (-0.870 + 0.491i)T \) |
| 23 | \( 1 + (0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.198 - 0.980i)T \) |
| 31 | \( 1 + (-0.897 + 0.441i)T \) |
| 37 | \( 1 + (-0.696 - 0.717i)T \) |
| 41 | \( 1 + (0.0855 + 0.996i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 + (0.974 - 0.226i)T \) |
| 53 | \( 1 + (0.564 + 0.825i)T \) |
| 59 | \( 1 + (-0.0855 + 0.996i)T \) |
| 61 | \( 1 + (0.516 + 0.856i)T \) |
| 67 | \( 1 + (0.654 + 0.755i)T \) |
| 71 | \( 1 + (-0.736 - 0.676i)T \) |
| 73 | \( 1 + (-0.941 + 0.336i)T \) |
| 79 | \( 1 + (0.254 - 0.967i)T \) |
| 83 | \( 1 + (0.0285 - 0.999i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (0.774 + 0.633i)T \) |
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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
−18.36440056554105682423892242586, −17.76341490378484684532402283666, −17.10489986696824141349626724310, −16.64584426348115648871583467765, −15.751097148848704598093411300, −15.56093804489667335108283330596, −14.74661754063149444181835030572, −14.066764269441006934930870204579, −13.070779221708017343176343790988, −12.80169515851233145081966421880, −11.87558205522822669694337707737, −11.00328247074557655034930432966, −10.78946516911459057837197593789, −9.487597259010785465620254275588, −8.9248729594253240312552067280, −8.387814271986262834373895757517, −7.11356176495752562722380252731, −6.74564713497230621264603735292, −6.02353152483096716419850808372, −5.29362347219038378885319555252, −4.657938477030828360553802395272, −3.95731207194973912437726680704, −3.33631668077672939556133891, −2.18233338281339946764118929220, −0.77019692691824169230803698628,
0.44082397060663188586447702247, 1.4034564883636430995338274217, 2.048466988626133976176585167666, 2.91516100369460502059533593046, 4.0228955246024214080468220014, 4.44868240453674833341889049136, 5.57532625018686309687493697884, 5.873692492982719803039983097793, 6.70766825661776351646321563917, 7.47048744176219243204149557783, 8.62736969754256109336784538530, 9.0531352035967115684568911314, 10.29718950194016473178832565541, 10.68583102487045467343619510056, 11.28907761251730637492209640326, 11.9027172434968437551438938915, 12.69265898521768059708367811562, 13.19064467307647530628810721252, 13.63711372853964771178042612448, 14.55912131717868177233294741891, 15.34013537625099657872027961366, 15.9857400395711919555493190432, 16.898963542884085102990972146, 17.63355263807283953662739933839, 18.12161345492523463909160435106