L(s) = 1 | + (0.433 − 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 − 0.433i)6-s + (−0.623 + 0.781i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.974 + 0.222i)11-s − i·12-s + (−0.222 + 0.974i)13-s + (0.433 + 0.900i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.974 + 0.222i)18-s + (0.781 − 0.623i)19-s + ⋯ |
L(s) = 1 | + (0.433 − 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 − 0.433i)6-s + (−0.623 + 0.781i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.974 + 0.222i)11-s − i·12-s + (−0.222 + 0.974i)13-s + (0.433 + 0.900i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.974 + 0.222i)18-s + (0.781 − 0.623i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.308662544 + 0.6395508403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.308662544 + 0.6395508403i\) |
\(L(1)\) |
\(\approx\) |
\(1.532915203 - 0.05958206186i\) |
\(L(1)\) |
\(\approx\) |
\(1.532915203 - 0.05958206186i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.433 - 0.900i)T \) |
| 3 | \( 1 + (0.781 + 0.623i)T \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.974 + 0.222i)T \) |
| 13 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.781 - 0.623i)T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 31 | \( 1 + (-0.433 + 0.900i)T \) |
| 37 | \( 1 + (-0.974 + 0.222i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.433 - 0.900i)T \) |
| 47 | \( 1 + (0.974 + 0.222i)T \) |
| 53 | \( 1 + (0.900 + 0.433i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.781 - 0.623i)T \) |
| 67 | \( 1 + (-0.222 - 0.974i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.433 + 0.900i)T \) |
| 79 | \( 1 + (-0.974 + 0.222i)T \) |
| 83 | \( 1 + (-0.623 - 0.781i)T \) |
| 89 | \( 1 + (-0.433 + 0.900i)T \) |
| 97 | \( 1 + (0.781 - 0.623i)T \) |
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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
−27.37302509378858530559120920833, −26.74173038907850771185753460816, −25.65211969469222518527973828457, −24.96465370173894673229884420236, −24.22199638024947903683918432863, −22.99252729545546555269706716013, −22.426556916304675769569893637253, −20.85710982204754086556324796629, −19.96996891830226293947665851080, −18.850757632501686897411242772665, −17.72160819295942159632637464498, −16.755658986984237026216658143425, −15.62139588575917029662470753487, −14.52559775563642419090922574348, −13.726438541887909364139297738831, −12.93770026342054905927104686223, −11.83471850155406449529403769102, −9.80454834408405860429381954382, −8.84197415753241847353531750964, −7.52138743540521720187229965344, −6.95348076145338541709340152991, −5.632498539655283567590535181622, −3.90671968658797435355183595360, −3.03635548552315046083341762691, −0.76221965105664081328848503788,
1.73360626991827547458521890467, 2.97395581878855134724016550417, 3.99794313283905971488734067016, 5.175130304078240554638390675534, 6.70454251910651342254803175217, 8.806715469540763474932988940942, 9.26897522962582279825041771155, 10.37936037723923160869018508517, 11.643401467716270427371800262346, 12.66379114327271944003991461242, 13.82124045122886456653688031935, 14.71679364185535726537936584618, 15.567579214780022233649307916907, 16.93592050516019844392455058235, 18.571872845722576194810106623563, 19.41593060014395156794152281014, 20.027786426535186414221857673283, 21.31610041586097438664639277557, 21.87986695188661651783546416889, 22.692219794764651069960378703197, 24.14707390925854144457440103390, 25.14706816541733541713380906312, 26.286240009188268168689481972582, 27.21107410706683633604442738276, 28.252453534159873365582205773983