L(s) = 1 | + (0.882 + 0.469i)2-s + (−0.997 − 0.0697i)3-s + (0.559 + 0.829i)4-s + (−0.615 − 0.788i)5-s + (−0.848 − 0.529i)6-s + (0.104 − 0.994i)7-s + (0.104 + 0.994i)8-s + (0.990 + 0.139i)9-s + (−0.173 − 0.984i)10-s + (−0.5 − 0.866i)12-s + (0.374 + 0.927i)13-s + (0.559 − 0.829i)14-s + (0.559 + 0.829i)15-s + (−0.374 + 0.927i)16-s + (−0.990 + 0.139i)17-s + (0.809 + 0.587i)18-s + ⋯ |
L(s) = 1 | + (0.882 + 0.469i)2-s + (−0.997 − 0.0697i)3-s + (0.559 + 0.829i)4-s + (−0.615 − 0.788i)5-s + (−0.848 − 0.529i)6-s + (0.104 − 0.994i)7-s + (0.104 + 0.994i)8-s + (0.990 + 0.139i)9-s + (−0.173 − 0.984i)10-s + (−0.5 − 0.866i)12-s + (0.374 + 0.927i)13-s + (0.559 − 0.829i)14-s + (0.559 + 0.829i)15-s + (−0.374 + 0.927i)16-s + (−0.990 + 0.139i)17-s + (0.809 + 0.587i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02672731429 + 0.4443552321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02672731429 + 0.4443552321i\) |
\(L(1)\) |
\(\approx\) |
\(0.9178281649 + 0.2091179977i\) |
\(L(1)\) |
\(\approx\) |
\(0.9178281649 + 0.2091179977i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.882 + 0.469i)T \) |
| 3 | \( 1 + (-0.997 - 0.0697i)T \) |
| 5 | \( 1 + (-0.615 - 0.788i)T \) |
| 7 | \( 1 + (0.104 - 0.994i)T \) |
| 13 | \( 1 + (0.374 + 0.927i)T \) |
| 17 | \( 1 + (-0.990 + 0.139i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.438 + 0.898i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.997 + 0.0697i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.719 - 0.694i)T \) |
| 53 | \( 1 + (-0.615 + 0.788i)T \) |
| 59 | \( 1 + (-0.719 + 0.694i)T \) |
| 61 | \( 1 + (-0.0348 + 0.999i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.615 - 0.788i)T \) |
| 73 | \( 1 + (0.241 + 0.970i)T \) |
| 79 | \( 1 + (-0.848 + 0.529i)T \) |
| 83 | \( 1 + (-0.669 + 0.743i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.882 - 0.469i)T \) |
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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
−26.00423714801372512458180012893, −24.66943543320751544091082734119, −23.96997184722631054350591328339, −22.87903391769799342236450492303, −22.38170673105661967860109696838, −21.717746064171809877795989913, −20.57617520191343220603416195111, −19.37208791366555920304754362400, −18.480636746020974309674326527179, −17.70337084041329232310202524390, −15.84321644437322457061254031940, −15.59747472000214883417641379483, −14.53502229664230638880964808, −13.123608430200665539413900527422, −12.235367280081726441236020669, −11.35916070600102434742685105488, −10.80125136354735211746730282505, −9.596594891312039989787345328968, −7.74193658486161387049472937288, −6.426985726903979753345186235076, −5.73253927789901494310712793387, −4.54190113014829086482869692269, −3.36707491646938077196414437741, −2.020938279225062687197887251441, −0.12198113238438133508437423169,
1.60788433164578216603746888352, 3.9557169435417077439981231816, 4.40978345821659264453468381702, 5.58929692181022948231718450138, 6.7908770467831463248209423380, 7.55003492209996098933690001010, 8.90662501630956153351688052086, 10.728417698661676204503820005831, 11.461440663973590607414315226773, 12.45720774626457486909420848263, 13.22844379808036793588332007796, 14.29307657567466793429799520164, 15.73486832971985278278994346949, 16.33560097836980150138226142273, 17.01701856127888299634945586935, 18.01959340787283889955998354069, 19.623807828424670233579901083976, 20.53705866557127423357225222132, 21.48540424557955785934153759276, 22.504849984562353601921488068698, 23.34309485457739338757445413960, 24.03261794016562947865069436510, 24.43160914891580341060270383913, 26.01307140819782962711440135290, 26.8701529357573424831848790422