L(s) = 1 | + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + i·7-s + 8-s + 9-s + i·10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + i·7-s + 8-s + 9-s + i·10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3634052676 + 0.1088154495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3634052676 + 0.1088154495i\) |
\(L(1)\) |
\(\approx\) |
\(0.4975545579 - 0.04754349274i\) |
\(L(1)\) |
\(\approx\) |
\(0.4975545579 - 0.04754349274i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-0.866 + 0.5i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (0.866 - 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−31.75591268486242532889607670117, −30.10320967234611649667614896711, −29.268643037651605666410315174, −27.72659948456073073857654787683, −27.18255852406004615616852687229, −26.38395028749233752314146823668, −24.6697782337390974475259061816, −23.849732458159579136883326626378, −22.85809432601192758401448738467, −22.21366601782945139821030804459, −20.00079433439226378220151546729, −19.04514913678649974828389564370, −17.79760986360941965867560329791, −16.90621862680540176633241762365, −15.97921063394072991325160071591, −14.83833770372785792476043327013, −13.47952931040345334401120005572, −11.6436723283997434740220882432, −10.72805553287902299362619030997, −9.43408077137460782326451155658, −7.42807885054769575824129548669, −7.02147709981753888683179453363, −5.37994343923226028325369470933, −4.01316532047686286463052181923, −0.631266945954063859501442755,
1.652654487231475692726238820130, 3.86757530674488778985387704566, 5.11110852365547772255347322085, 7.0541629301486713284230608127, 8.569943113509999250688701538528, 9.76883265106743977249876170764, 11.222576678292564016862689243382, 12.19631130077201137257770283609, 12.57328654819270183231195387237, 14.90176027592689337679870607449, 16.41247267379965200207444899086, 17.151296829020622598240301118078, 18.50679429529993773199952815650, 19.3096946371100196999653672989, 20.579248118492433735081946492600, 21.90237500185971694409361826650, 22.53178671189727644810117938046, 23.904726856236958691441773727994, 25.08759742332981502015799124775, 26.83306005827238116018222462464, 27.63303737182188125150935188633, 28.37806407008272789283606388720, 29.151646535034788535754334509609, 30.507636899704489275372694374671, 31.33860501829332766570581209489