L(s) = 1 | + (−0.425 + 0.904i)2-s + (0.876 + 0.481i)3-s + (−0.637 − 0.770i)4-s + (−0.425 − 0.904i)5-s + (−0.809 + 0.587i)6-s + (0.728 − 0.684i)7-s + (0.968 − 0.248i)8-s + (0.535 + 0.844i)9-s + 10-s + (0.535 + 0.844i)11-s + (−0.187 − 0.982i)12-s + (0.728 + 0.684i)13-s + (0.309 + 0.951i)14-s + (0.0627 − 0.998i)15-s + (−0.187 + 0.982i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
L(s) = 1 | + (−0.425 + 0.904i)2-s + (0.876 + 0.481i)3-s + (−0.637 − 0.770i)4-s + (−0.425 − 0.904i)5-s + (−0.809 + 0.587i)6-s + (0.728 − 0.684i)7-s + (0.968 − 0.248i)8-s + (0.535 + 0.844i)9-s + 10-s + (0.535 + 0.844i)11-s + (−0.187 − 0.982i)12-s + (0.728 + 0.684i)13-s + (0.309 + 0.951i)14-s + (0.0627 − 0.998i)15-s + (−0.187 + 0.982i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9551649212 + 0.4396264903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9551649212 + 0.4396264903i\) |
\(L(1)\) |
\(\approx\) |
\(0.9959773437 + 0.3722989974i\) |
\(L(1)\) |
\(\approx\) |
\(0.9959773437 + 0.3722989974i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 101 | \( 1 \) |
good | 2 | \( 1 + (-0.425 + 0.904i)T \) |
| 3 | \( 1 + (0.876 + 0.481i)T \) |
| 5 | \( 1 + (-0.425 - 0.904i)T \) |
| 7 | \( 1 + (0.728 - 0.684i)T \) |
| 11 | \( 1 + (0.535 + 0.844i)T \) |
| 13 | \( 1 + (0.728 + 0.684i)T \) |
| 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.187 - 0.982i)T \) |
| 23 | \( 1 + (-0.992 - 0.125i)T \) |
| 29 | \( 1 + (0.728 + 0.684i)T \) |
| 31 | \( 1 + (0.728 - 0.684i)T \) |
| 37 | \( 1 + (0.876 - 0.481i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (-0.929 + 0.368i)T \) |
| 47 | \( 1 + (-0.929 - 0.368i)T \) |
| 53 | \( 1 + (-0.637 + 0.770i)T \) |
| 59 | \( 1 + (-0.187 + 0.982i)T \) |
| 61 | \( 1 + (-0.637 - 0.770i)T \) |
| 67 | \( 1 + (0.876 - 0.481i)T \) |
| 71 | \( 1 + (0.876 + 0.481i)T \) |
| 73 | \( 1 + (-0.992 - 0.125i)T \) |
| 79 | \( 1 + (-0.992 + 0.125i)T \) |
| 83 | \( 1 + (-0.992 + 0.125i)T \) |
| 89 | \( 1 + (-0.187 - 0.982i)T \) |
| 97 | \( 1 + (-0.637 - 0.770i)T \) |
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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
−30.20511843173638336231640099022, −28.86052878166721126862312736390, −27.46974014342236221757243663821, −26.884905032531207898727716911672, −25.8094008123298297379647885221, −24.82773248157988507915864072880, −23.46584603063015266575305056218, −22.127041907157318573776113943125, −21.25279107500101208592164336280, −20.11204020560807005115911284747, −19.17204791292739327718313410549, −18.44257011198151666576587178379, −17.6343316115120205825796879238, −15.68816005757535940912843722344, −14.48650507056526841706208356304, −13.5720010275356055028477872166, −12.184686682645804013326380609984, −11.27629173914567358197252331560, −10.05655046474211492749576709144, −8.400084864653265839138473945498, −8.15581414246114727331408521307, −6.35018477905551529959052890008, −3.97879125254521148782511952395, −2.96044529334050869108680335941, −1.67398462177915119821629795452,
1.57719277422445822209631331114, 4.27187918063757632404081594817, 4.69865037997089211833961795222, 6.84489338033224297539241451118, 8.0136769268281372915777791525, 8.84769707764340526070832111275, 9.81411252034858673190189055289, 11.31384180456045251046111929559, 13.2595214350267231189368574061, 14.11108002364492683850428327618, 15.23265801367492699300380250278, 16.10377929292805425131602618274, 17.0817145278762118827706513623, 18.26137630202024561569179362795, 19.872119309848946424534023279657, 20.14741324287322152693149557380, 21.5863002204447955525452098807, 23.134403876452488077283685689, 24.117758068580405505144385155157, 24.88377648457206429978587428535, 25.98747454339192063217466802445, 26.83915969452806588010868515851, 27.74715965563351308493690122015, 28.426026827921583339276841886251, 30.437413268020117471706322444081