L(s) = 1 | + (0.195 + 0.980i)2-s + (0.382 + 0.923i)3-s + (−0.923 + 0.382i)4-s + (−0.634 + 0.773i)5-s + (−0.831 + 0.555i)6-s + (−0.707 + 0.707i)7-s + (−0.555 − 0.831i)8-s + (−0.707 + 0.707i)9-s + (−0.881 − 0.471i)10-s + (0.0980 − 0.995i)11-s + (−0.707 − 0.707i)12-s + (−0.471 − 0.881i)13-s + (−0.831 − 0.555i)14-s + (−0.956 − 0.290i)15-s + (0.707 − 0.707i)16-s + (0.634 + 0.773i)17-s + ⋯ |
L(s) = 1 | + (0.195 + 0.980i)2-s + (0.382 + 0.923i)3-s + (−0.923 + 0.382i)4-s + (−0.634 + 0.773i)5-s + (−0.831 + 0.555i)6-s + (−0.707 + 0.707i)7-s + (−0.555 − 0.831i)8-s + (−0.707 + 0.707i)9-s + (−0.881 − 0.471i)10-s + (0.0980 − 0.995i)11-s + (−0.707 − 0.707i)12-s + (−0.471 − 0.881i)13-s + (−0.831 − 0.555i)14-s + (−0.956 − 0.290i)15-s + (0.707 − 0.707i)16-s + (0.634 + 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4250062101 + 0.2048140768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4250062101 + 0.2048140768i\) |
\(L(1)\) |
\(\approx\) |
\(0.3917975778 + 0.7050198852i\) |
\(L(1)\) |
\(\approx\) |
\(0.3917975778 + 0.7050198852i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 193 | \( 1 \) |
good | 2 | \( 1 + (0.195 + 0.980i)T \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (-0.634 + 0.773i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.0980 - 0.995i)T \) |
| 13 | \( 1 + (-0.471 - 0.881i)T \) |
| 17 | \( 1 + (0.634 + 0.773i)T \) |
| 19 | \( 1 + (0.773 + 0.634i)T \) |
| 23 | \( 1 + (0.195 + 0.980i)T \) |
| 29 | \( 1 + (0.290 + 0.956i)T \) |
| 31 | \( 1 + (-0.980 + 0.195i)T \) |
| 37 | \( 1 + (0.290 - 0.956i)T \) |
| 41 | \( 1 + (-0.0980 + 0.995i)T \) |
| 43 | \( 1 + (-0.707 - 0.707i)T \) |
| 47 | \( 1 + (-0.881 + 0.471i)T \) |
| 53 | \( 1 + (-0.471 - 0.881i)T \) |
| 59 | \( 1 + (0.382 + 0.923i)T \) |
| 61 | \( 1 + (0.773 - 0.634i)T \) |
| 67 | \( 1 + (-0.980 + 0.195i)T \) |
| 71 | \( 1 + (-0.773 - 0.634i)T \) |
| 73 | \( 1 + (-0.290 - 0.956i)T \) |
| 79 | \( 1 + (0.995 + 0.0980i)T \) |
| 83 | \( 1 + (0.555 - 0.831i)T \) |
| 89 | \( 1 + (0.471 - 0.881i)T \) |
| 97 | \( 1 + (0.195 - 0.980i)T \) |
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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
−26.15953186785737641233844165961, −24.87197794599962052953505395691, −23.83733437199961204810935226044, −23.20555276327165452888519227737, −22.3582460687068570466876419460, −20.6845964666645843636330315643, −20.24421303306749503101517131620, −19.4412019998108813603997766564, −18.68960306068743439469807598870, −17.49752990566683621548484946334, −16.47881417726309408563208343869, −14.90931713135423166093389909092, −13.84516379074241262693004079068, −13.01653327344440393677059574897, −12.19897359770921263257611770094, −11.52275035698351791857762778386, −9.82469818142747536501465795095, −9.11770623291577909736499966583, −7.78545931539664737222658162803, −6.78024754458156231732415410856, −4.99350658019211998672679610075, −3.91767125688644583013746295602, −2.66593792041859242937012299960, −1.26460750894296576089580850783, −0.159521108856181204606122544290,
3.25137692925504024004246335583, 3.478409975654738982927377875026, 5.24412007147647505634855673714, 6.054534068239987485472638527201, 7.542279417179362581422011132606, 8.40787276928123247425645438370, 9.500511121853854251577508841689, 10.509184733107185918029192818987, 11.90365143670906077295998861606, 13.17978918755699444114948278499, 14.48940175998725447210911522681, 14.94286824690654132131626332384, 15.994239409044864321889026544921, 16.43706970196738324064191401460, 17.915764238609895524261813718627, 19.00374103339076494171341700496, 19.75529566250359526839524249190, 21.442149181048062091125216661054, 22.06626298606192698412907036590, 22.742139879386093931995060510096, 23.77299970685322696153212734267, 25.1296894615299657795651547590, 25.6418156834672757289427103377, 26.71449342128310790370889286226, 27.19811713004267570547121657862