L(s) = 1 | + (0.0475 − 0.998i)2-s + (−0.786 + 0.618i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (0.580 + 0.814i)6-s + (0.928 − 0.371i)7-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.723 + 0.690i)10-s + (−0.959 + 0.281i)11-s + (0.841 − 0.540i)12-s + (0.0475 − 0.998i)13-s + (−0.327 − 0.945i)14-s + (0.0475 − 0.998i)15-s + (0.981 + 0.189i)16-s + (−0.959 + 0.281i)17-s + ⋯ |
L(s) = 1 | + (0.0475 − 0.998i)2-s + (−0.786 + 0.618i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (0.580 + 0.814i)6-s + (0.928 − 0.371i)7-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.723 + 0.690i)10-s + (−0.959 + 0.281i)11-s + (0.841 − 0.540i)12-s + (0.0475 − 0.998i)13-s + (−0.327 − 0.945i)14-s + (0.0475 − 0.998i)15-s + (0.981 + 0.189i)16-s + (−0.959 + 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.007286800617 - 0.2316629505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.007286800617 - 0.2316629505i\) |
\(L(1)\) |
\(\approx\) |
\(0.4988042730 - 0.1986897381i\) |
\(L(1)\) |
\(\approx\) |
\(0.4988042730 - 0.1986897381i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (0.0475 - 0.998i)T \) |
| 3 | \( 1 + (-0.786 + 0.618i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.928 - 0.371i)T \) |
| 11 | \( 1 + (-0.959 + 0.281i)T \) |
| 13 | \( 1 + (0.0475 - 0.998i)T \) |
| 17 | \( 1 + (-0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.995 + 0.0950i)T \) |
| 29 | \( 1 + (-0.888 - 0.458i)T \) |
| 31 | \( 1 + (-0.786 - 0.618i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.235 - 0.971i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.928 - 0.371i)T \) |
| 53 | \( 1 + (0.0475 + 0.998i)T \) |
| 59 | \( 1 + (0.415 - 0.909i)T \) |
| 61 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (-0.327 + 0.945i)T \) |
| 73 | \( 1 + (-0.995 + 0.0950i)T \) |
| 79 | \( 1 + (0.981 + 0.189i)T \) |
| 83 | \( 1 + (0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.888 - 0.458i)T \) |
| 97 | \( 1 + (-0.786 - 0.618i)T \) |
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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
−27.405119942805727687732610240, −26.48206190311320011701369848576, −25.14501459744982946780863021573, −24.18199522531876629234544037324, −23.95475304238659332856960673898, −23.09813091550928059879495525277, −21.92186946150575181241018408645, −20.97159073216779693484085030248, −19.41697989613918606844567682363, −18.374660185547397301417053555607, −17.825172835200787645959705319453, −16.525023500545488455735761234889, −16.16480701570845912974145185545, −14.91904277815374482390480830170, −13.72792085688357089456986711184, −12.75781004631641866595343836805, −11.87027054562117918156464602307, −10.77810932988968641086203944308, −8.99789353256024159704437011604, −8.09502656878146489162406741440, −7.33292195488052963805641920800, −5.99165799175117065992756419049, −5.06634745958928444934522891869, −4.226082338287357105026827037224, −1.71558647385874264955081079135,
0.19209851887449814057143802617, 2.270358218405334886932048520879, 3.723964988894071098190088070519, 4.584982160483613411423445252229, 5.66896059057706527543073060941, 7.390143173910375670316333888976, 8.551563887733658407402868060188, 10.11213536840288067346806476994, 10.79017549610174417825862645404, 11.32283246231390463922551664969, 12.417326242243872063607695710606, 13.58622202419621926907883543611, 15.00252912348132114655572279241, 15.4977490581616667643776792193, 17.23863115668233798219103556504, 17.89252745113905212861533028932, 18.67940314098275973765915663070, 20.1346619578460826875496426473, 20.643455200111442911830454891924, 21.91105671654846286209619342647, 22.368458209366640054280248086155, 23.53225382775561399159090277160, 23.891067733305025895903298996524, 26.161451482334642268377616111202, 26.65951438626405543495793117652