L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (0.669 − 0.743i)11-s + (0.913 − 0.406i)12-s + (0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.669 + 0.743i)17-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (0.669 − 0.743i)11-s + (0.913 − 0.406i)12-s + (0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.669 + 0.743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1991632166 - 0.4010864112i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1991632166 - 0.4010864112i\) |
\(L(1)\) |
\(\approx\) |
\(0.4515293016 - 0.3943007268i\) |
\(L(1)\) |
\(\approx\) |
\(0.4515293016 - 0.3943007268i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.978 - 0.207i)T \) |
| 11 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (0.913 + 0.406i)T \) |
| 17 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.104 + 0.994i)T \) |
| 43 | \( 1 + (0.913 - 0.406i)T \) |
| 47 | \( 1 + (-0.809 + 0.587i)T \) |
| 53 | \( 1 + (-0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.978 + 0.207i)T \) |
| 73 | \( 1 + (0.669 - 0.743i)T \) |
| 79 | \( 1 + (0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.104 + 0.994i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−37.56117296713016948724792621859, −35.689963066360949574821666417, −34.86373753237590016340876449647, −33.60422829030002860036916166493, −32.787294138571927053994561742077, −31.41962530416648618740826247448, −29.52573433486529770880510115484, −28.09289757587272299545185740700, −27.28292481980241567720003971917, −26.06529086873843906755747371673, −25.3396040787606141805274766918, −23.11179038120913312256880680141, −22.50034671831469411927423953910, −20.44722826501447489017838011460, −19.24276256119616348481805989425, −17.889019659470139420285862721115, −16.29133423944569065859467695929, −15.508707777237651210436166139754, −14.32133379179052035490221336546, −11.57167656714932540999543295535, −10.21279093390690613521337702032, −9.18645579116628708017627918036, −7.27314917182395729881768357459, −5.74284126321691369338654880447, −3.4187048448422688456194145716,
1.100148492494785447431532988451, 3.47167474507840179793151472467, 6.4276988293955424104253576982, 8.03689453383942110138629714492, 9.201544748730044035377593896283, 11.26009480707037679870443917191, 12.412325838087355522215203576311, 13.46073606637374061359398965025, 16.21817782138821658404278705961, 17.04610307959612534830308096495, 18.76370345171913444247532723068, 19.478001555937092583680958193664, 20.63455765387678887955098675592, 22.48393053556971025409800272981, 23.99568726583195979097293085612, 25.20586496146279754310082231991, 26.44181962703953329506332515199, 28.08991154244444441648225340610, 28.83638488602228454958944700281, 30.00360286960329453477082336644, 31.10084512061844517431714857394, 32.53836983556514297342012883642, 34.731546842609679255158578041822, 35.4072580441796122382534135979, 36.17544753793386369783800007059