L(s) = 1 | + (0.347 + 0.937i)2-s + (0.347 − 0.937i)3-s + (−0.758 + 0.651i)4-s + (−0.954 + 0.299i)5-s + 6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.758 − 0.651i)9-s + (−0.612 − 0.790i)10-s + (−0.820 − 0.571i)11-s + (0.347 + 0.937i)12-s + (−0.758 − 0.651i)13-s + (−0.994 − 0.101i)14-s + (−0.0506 + 0.998i)15-s + (0.151 − 0.988i)16-s + (0.612 + 0.790i)17-s + ⋯ |
L(s) = 1 | + (0.347 + 0.937i)2-s + (0.347 − 0.937i)3-s + (−0.758 + 0.651i)4-s + (−0.954 + 0.299i)5-s + 6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.758 − 0.651i)9-s + (−0.612 − 0.790i)10-s + (−0.820 − 0.571i)11-s + (0.347 + 0.937i)12-s + (−0.758 − 0.651i)13-s + (−0.994 − 0.101i)14-s + (−0.0506 + 0.998i)15-s + (0.151 − 0.988i)16-s + (0.612 + 0.790i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.328822652 + 0.1595775922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328822652 + 0.1595775922i\) |
\(L(1)\) |
\(\approx\) |
\(0.9304564387 + 0.2501864737i\) |
\(L(1)\) |
\(\approx\) |
\(0.9304564387 + 0.2501864737i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (0.347 + 0.937i)T \) |
| 3 | \( 1 + (0.347 - 0.937i)T \) |
| 5 | \( 1 + (-0.954 + 0.299i)T \) |
| 7 | \( 1 + (-0.440 + 0.897i)T \) |
| 11 | \( 1 + (-0.820 - 0.571i)T \) |
| 13 | \( 1 + (-0.758 - 0.651i)T \) |
| 17 | \( 1 + (0.612 + 0.790i)T \) |
| 19 | \( 1 + (0.954 + 0.299i)T \) |
| 23 | \( 1 + (0.874 + 0.485i)T \) |
| 29 | \( 1 + (0.994 - 0.101i)T \) |
| 31 | \( 1 + (0.612 - 0.790i)T \) |
| 37 | \( 1 + (0.440 + 0.897i)T \) |
| 41 | \( 1 + (-0.918 - 0.394i)T \) |
| 43 | \( 1 + (0.440 - 0.897i)T \) |
| 47 | \( 1 + (-0.994 + 0.101i)T \) |
| 53 | \( 1 + (-0.440 + 0.897i)T \) |
| 59 | \( 1 + (0.440 - 0.897i)T \) |
| 61 | \( 1 + (0.954 + 0.299i)T \) |
| 67 | \( 1 + (0.918 - 0.394i)T \) |
| 71 | \( 1 + (-0.979 - 0.201i)T \) |
| 73 | \( 1 + (0.528 + 0.848i)T \) |
| 79 | \( 1 + (0.918 + 0.394i)T \) |
| 83 | \( 1 + (-0.0506 - 0.998i)T \) |
| 89 | \( 1 + (-0.440 - 0.897i)T \) |
| 97 | \( 1 + (-0.979 - 0.201i)T \) |
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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
−24.93708670281040341132986484501, −23.62937620888386227088631649935, −23.040699820031964929673386687, −22.33912040396256575037856008254, −21.10517774670626270592670849740, −20.59186936130466266244939180261, −19.72038841663835414926155955957, −19.261007703828698375276958265600, −17.88594771006873611128199246844, −16.56104561365296152647704233316, −15.86808056104803619545110128715, −14.7811630722604782517007226931, −13.999439172983191786983902963933, −12.94316308828735930630218922871, −11.91793443698367554731775597513, −11.03793418928896755931021345481, −10.05712139114714378275615746282, −9.451376988974951943070086620274, −8.21364139193450686188203547806, −7.069123840496939415689857089069, −4.98208612261417304635930180794, −4.6679605908668995691330144727, −3.472034184861640279694384420276, −2.71110718830337096671519289844, −0.75238193880979355690292850287,
0.52782498080365812244817285555, 2.82071807094832301273231881794, 3.3779639589378150276683694447, 5.13763380812926581795032706400, 6.03558208012880741646152413991, 7.10628011164649377906791757777, 7.98966204232718479322309158432, 8.47185061853202277019846811240, 9.85658303191781345160416184711, 11.617305743725656849922712467878, 12.38406505915313077530117594475, 13.086576236980750822602577493218, 14.16691153700122285457843459885, 15.16341646071326725754462927569, 15.61395894360997479586493324654, 16.82065586791992466000575540157, 17.89801864248956572099405862445, 18.836610770204343444381333784004, 19.18823397947488206485741859149, 20.54873481032781196715539072463, 21.80666411183188146267088236175, 22.68263686028236011847219796648, 23.463590607854487607557641786805, 24.170379144211516141210523195717, 24.96794544091862613686218904331