L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.866 − 0.5i)3-s + (0.173 + 0.984i)4-s + (−0.0871 + 0.996i)5-s + (−0.342 − 0.939i)6-s + (−0.258 − 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (−0.707 + 0.707i)10-s + (−0.996 − 0.0871i)11-s + (0.342 − 0.939i)12-s + (−0.906 − 0.422i)13-s + (0.422 − 0.906i)14-s + (0.573 − 0.819i)15-s + (−0.939 + 0.342i)16-s + (−0.965 − 0.258i)17-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.866 − 0.5i)3-s + (0.173 + 0.984i)4-s + (−0.0871 + 0.996i)5-s + (−0.342 − 0.939i)6-s + (−0.258 − 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (−0.707 + 0.707i)10-s + (−0.996 − 0.0871i)11-s + (0.342 − 0.939i)12-s + (−0.906 − 0.422i)13-s + (0.422 − 0.906i)14-s + (0.573 − 0.819i)15-s + (−0.939 + 0.342i)16-s + (−0.965 − 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05889098187 + 0.5984746350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05889098187 + 0.5984746350i\) |
\(L(1)\) |
\(\approx\) |
\(0.7398091698 + 0.4146875989i\) |
\(L(1)\) |
\(\approx\) |
\(0.7398091698 + 0.4146875989i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.0871 + 0.996i)T \) |
| 7 | \( 1 + (-0.258 - 0.965i)T \) |
| 11 | \( 1 + (-0.996 - 0.0871i)T \) |
| 13 | \( 1 + (-0.906 - 0.422i)T \) |
| 17 | \( 1 + (-0.965 - 0.258i)T \) |
| 19 | \( 1 + (-0.642 + 0.766i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (0.0871 + 0.996i)T \) |
| 31 | \( 1 + (0.819 - 0.573i)T \) |
| 37 | \( 1 + (0.766 - 0.642i)T \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (-0.965 + 0.258i)T \) |
| 47 | \( 1 + (0.422 + 0.906i)T \) |
| 53 | \( 1 + (0.996 - 0.0871i)T \) |
| 59 | \( 1 + (-0.422 + 0.906i)T \) |
| 61 | \( 1 + (-0.342 + 0.939i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (-0.342 - 0.939i)T \) |
| 83 | \( 1 + (-0.707 + 0.707i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−30.916818956081061406516845149101, −29.24740992469502995528843114225, −28.59713820998148938143912671725, −28.08578918264171113586769604216, −26.69692401149675573663576419777, −24.76726923264273354100155407081, −23.952052741015550137619063809, −22.91832779499374187295612226044, −21.71225789723522197844866104543, −21.200884898979956893910593916, −19.899752695549674506125881243082, −18.6120695774961763547233718386, −17.18329426297742907912970191768, −15.78632179341797054208063022304, −15.15964009986707254919213915488, −13.14578246458143573857473927659, −12.381725524962284183378423914932, −11.40492308160944975155130065909, −10.06407121618241840300623010486, −8.91546993913225702539725761500, −6.44883615456423166708711395303, −5.15872618726447535317887182891, −4.47525091883086052503508103220, −2.40623350299029884134848122441, −0.24084317049798556040571879378,
2.70592218842934112239345678813, 4.44571826715134159247872992334, 5.873453270259552784845750513283, 7.04783720094396163504038888077, 7.71026082912633528011083392133, 10.34906542203593032445882218063, 11.32178458785871140375410154571, 12.76738913308966442434913819453, 13.61436880765387447417865809398, 14.947800094390657491844167453866, 16.15722442665181002147605423311, 17.282607323997517055124891457875, 18.16361873008462094189234316912, 19.63307153835664310279060491631, 21.320160925861302599178414732178, 22.44551679781077328023939278483, 23.11965439409703057347046028993, 23.90212719923291114329025580917, 25.15115004301265975769973436405, 26.38400893129936875148566463582, 27.19892124963294328469193151804, 29.26210198515709258276296967488, 29.65207619085348980682002527041, 30.74444065364061311085245889122, 31.80986078657165386607938536266