L(s) = 1 | + (0.983 − 0.178i)2-s + (−0.963 − 0.266i)3-s + (0.936 − 0.351i)4-s + (0.309 + 0.951i)5-s + (−0.995 − 0.0896i)6-s + (−0.473 + 0.880i)7-s + (0.858 − 0.512i)8-s + (0.858 + 0.512i)9-s + (0.473 + 0.880i)10-s + (0.393 + 0.919i)11-s + (−0.995 + 0.0896i)12-s + (0.393 − 0.919i)13-s + (−0.309 + 0.951i)14-s + (−0.0448 − 0.998i)15-s + (0.753 − 0.657i)16-s + (0.809 + 0.587i)17-s + ⋯ |
L(s) = 1 | + (0.983 − 0.178i)2-s + (−0.963 − 0.266i)3-s + (0.936 − 0.351i)4-s + (0.309 + 0.951i)5-s + (−0.995 − 0.0896i)6-s + (−0.473 + 0.880i)7-s + (0.858 − 0.512i)8-s + (0.858 + 0.512i)9-s + (0.473 + 0.880i)10-s + (0.393 + 0.919i)11-s + (−0.995 + 0.0896i)12-s + (0.393 − 0.919i)13-s + (−0.309 + 0.951i)14-s + (−0.0448 − 0.998i)15-s + (0.753 − 0.657i)16-s + (0.809 + 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.201544748 + 0.7093912126i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.201544748 + 0.7093912126i\) |
\(L(1)\) |
\(\approx\) |
\(1.575691748 + 0.1824923047i\) |
\(L(1)\) |
\(\approx\) |
\(1.575691748 + 0.1824923047i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (0.983 - 0.178i)T \) |
| 3 | \( 1 + (-0.963 - 0.266i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.473 + 0.880i)T \) |
| 11 | \( 1 + (0.393 + 0.919i)T \) |
| 13 | \( 1 + (0.393 - 0.919i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.0448 + 0.998i)T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 29 | \( 1 + (-0.550 + 0.834i)T \) |
| 31 | \( 1 + (-0.753 - 0.657i)T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (0.134 - 0.990i)T \) |
| 47 | \( 1 + (0.963 - 0.266i)T \) |
| 53 | \( 1 + (-0.936 - 0.351i)T \) |
| 59 | \( 1 + (0.995 - 0.0896i)T \) |
| 61 | \( 1 + (-0.473 - 0.880i)T \) |
| 67 | \( 1 + (-0.936 + 0.351i)T \) |
| 73 | \( 1 + (0.983 - 0.178i)T \) |
| 79 | \( 1 + (0.858 - 0.512i)T \) |
| 83 | \( 1 + (-0.995 + 0.0896i)T \) |
| 89 | \( 1 + (0.936 + 0.351i)T \) |
| 97 | \( 1 + (0.222 - 0.974i)T \) |
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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
−31.64808485484989310612455767358, −30.008872202673614255985639639085, −29.283114583371187503667265697144, −28.45038021337380711263565570774, −27.02128256396921381353676589811, −25.67034937849666540965326348452, −24.25831895323378110345504898261, −23.66912800169882828668692684603, −22.61534222263487729866071320426, −21.46323289768027216598477586148, −20.74040062085259732376456722778, −19.29080877263544118686861624015, −17.228421109420185333790026250386, −16.57884509612366381402956577446, −15.816105589983013694126857729916, −13.952766695057222930944894806723, −13.091392504509484978181184419425, −11.86479538202910738769242636415, −10.859951585871624007482335679467, −9.25865814676062859161796256111, −7.16641595605316905275879779659, −6.0037762423858644434793680567, −4.854886860479310613886476916083, −3.689651854437828427604741320251, −1.069311816838886668374499597319,
1.87218477724955325206865122422, 3.497413008505889064944366732927, 5.39987184028395301146978586003, 6.20219199770779884591142578483, 7.34492727407821989658317817724, 9.98751405815799636516972454973, 10.92713764408183662304224967613, 12.23917051496583162777410954865, 12.91854032686866053604362864125, 14.58565932911506439460302628376, 15.45309464203994398740527307125, 16.83434696194042310368189702963, 18.270479761940994354021562955361, 19.1537487568447110486972188799, 20.84311834268289213583403304155, 22.08284618747630762989485836682, 22.61382513768809092491771029268, 23.42230828899378980509756970301, 24.98710924417779366231140571729, 25.57153588542734292059530402007, 27.616213618283718219843780729717, 28.55347797062718379444467472415, 29.62057303995202433958770376742, 30.30419502619791140978282238024, 31.36204357596145764323162364481