L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 + 0.974i)6-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + 12-s + (−0.623 + 0.781i)13-s + (0.222 − 0.974i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.623 − 0.781i)18-s + (0.900 − 0.433i)19-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 + 0.974i)6-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + 12-s + (−0.623 + 0.781i)13-s + (0.222 − 0.974i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.623 − 0.781i)18-s + (0.900 − 0.433i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6732767000 - 0.2044851760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6732767000 - 0.2044851760i\) |
\(L(1)\) |
\(\approx\) |
\(0.6856720671 - 0.2539945700i\) |
\(L(1)\) |
\(\approx\) |
\(0.6856720671 - 0.2539945700i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 3 | \( 1 + (-0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
| 11 | \( 1 + (-0.623 + 0.781i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.900 - 0.433i)T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 31 | \( 1 + (0.222 + 0.974i)T \) |
| 37 | \( 1 + (0.623 + 0.781i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.623 - 0.781i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + (-0.623 - 0.781i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 + (-0.623 - 0.781i)T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.222 + 0.974i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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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
−27.99330809577246167539800321818, −27.1295064706915210747131775850, −26.736534080527678275904441105720, −25.31617255696348628854492093381, −24.17935854276037916256241242583, −23.60284700203732014503501909956, −22.629199408148168465932088556014, −21.64209985731103384615208639009, −20.56459624020609219252326038769, −18.90864606602204887449010183638, −17.94807193021053944524389743262, −17.1804264232159656729175962319, −16.33341637814805650140956290162, −15.351159667746898252980579726324, −14.363293777301548191628998488008, −13.16487121700505107216456151787, −11.721179720267724037386452756228, −10.527346408651972967391765018638, −9.68815160873421003421561109606, −8.09288328017664226138246134337, −7.28029942895758757328809976948, −5.630668169781022636686622574131, −5.21339161124937516516081047118, −3.74651415918692782345728241795, −0.9055911157888081242952812359,
1.34814451645547212719105024127, 2.54741192188497095640903627874, 4.60254728348738675975143182299, 5.27285739928059949109876701013, 7.15098228193428270870191622998, 8.212794839336001131854331698622, 9.709134559688918607280601016228, 10.69288518667594008418816124351, 11.838104034660492961772267879, 12.26541710744593750192387737207, 13.53974101659140627505051440207, 14.74872666145142681749381883664, 16.41877307417731875843081487500, 17.429606852756857845723956291237, 18.24430081285213963657460189888, 18.89659793269281191833242438046, 20.24531142103140437805498663497, 21.28488663717917146548670313589, 22.0442761175390510330083965085, 23.12582227427405506593859737409, 23.94164257365627372781953419992, 25.133897341921026109108943598, 26.57546373107717419949525113104, 27.46081520298464016742454758251, 28.48085387975856517124292627285