L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + 5-s + (−0.900 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)12-s + (0.900 − 0.433i)13-s − 14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s − 17-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + 5-s + (−0.900 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)12-s + (0.900 − 0.433i)13-s − 14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s − 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1972070546 + 2.058781326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1972070546 + 2.058781326i\) |
\(L(1)\) |
\(\approx\) |
\(0.8680580018 + 1.166397078i\) |
\(L(1)\) |
\(\approx\) |
\(0.8680580018 + 1.166397078i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 3 | \( 1 + (-0.222 + 0.974i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.222 - 0.974i)T \) |
| 23 | \( 1 + (-0.623 + 0.781i)T \) |
| 29 | \( 1 + (-0.222 + 0.974i)T \) |
| 31 | \( 1 + (0.900 - 0.433i)T \) |
| 37 | \( 1 + (0.623 + 0.781i)T \) |
| 41 | \( 1 + (0.900 - 0.433i)T \) |
| 43 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 + (0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.623 - 0.781i)T \) |
| 67 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 + (-0.900 + 0.433i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.222 - 0.974i)T \) |
| 97 | \( 1 + (0.900 + 0.433i)T \) |
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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
−30.4740242763657227437531673779, −29.92308515228040949683894683147, −29.015292657370460377135005141128, −28.32377017585941479931105704111, −26.54490090458320537503248442527, −25.126866738197530284180258451150, −24.239982408160402939288003434726, −23.012274175984549121579701723769, −22.284698681190603936040218653861, −20.93243336657778955273672735076, −19.801334330786671728133255921976, −18.828182113271504894135713802697, −17.70983923732700927504599300413, −16.458211029540854083795298650254, −14.26542402086377187189722919101, −13.63631743645000416551508757704, −12.758014651258259185221834201244, −11.42732699456737750319393664853, −10.266102221558276911629345400275, −8.84870390347033503388908561339, −6.56827645825172395877868623127, −5.99969209107435885876441727429, −3.992118952082728207695929834338, −2.24052602379609027195986771425, −0.9194916361959866971185707092,
2.844479329795411345057359419709, 4.3758449589365584601916384826, 5.74564247540401083367342405773, 6.49588263293365142053827598418, 8.7952543603818659182047614777, 9.52393969868064278083580892310, 11.27635635805190107829123624830, 12.74502041795833781680457491004, 13.919834339383711748589701622590, 15.17381890090065741468638378772, 15.93434151801702910859686607087, 17.19131101846306494488837971181, 17.982803881713327358833387238, 20.12757335214581586618642101880, 21.411497465816845825930581994862, 22.10590377659126537315854049693, 22.80765097998088281373202812608, 24.41876948537222584514565949669, 25.649953464503600901082490365679, 25.96726370133662019757184625228, 27.57449143018824730891455318454, 28.56370448661153930550089469816, 29.89660991666521248611187931775, 31.19446533344730640453923202651, 32.39272901250379606138549795266