L(s) = 1 | + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (0.900 − 0.433i)8-s + (−0.0747 − 0.997i)10-s + (−0.365 + 0.930i)11-s + (−0.623 − 0.781i)13-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.222 − 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.733 − 0.680i)26-s + ⋯ |
L(s) = 1 | + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (0.900 − 0.433i)8-s + (−0.0747 − 0.997i)10-s + (−0.365 + 0.930i)11-s + (−0.623 − 0.781i)13-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.222 − 0.974i)20-s + (−0.222 + 0.974i)22-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.733 − 0.680i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.801097697 - 0.7180074460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.801097697 - 0.7180074460i\) |
\(L(1)\) |
\(\approx\) |
\(1.715471220 - 0.4294862353i\) |
\(L(1)\) |
\(\approx\) |
\(1.715471220 - 0.4294862353i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.988 - 0.149i)T \) |
| 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 11 | \( 1 + (-0.365 + 0.930i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| 17 | \( 1 + (-0.733 + 0.680i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.733 + 0.680i)T \) |
| 29 | \( 1 + (0.222 + 0.974i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.955 + 0.294i)T \) |
| 41 | \( 1 + (-0.900 + 0.433i)T \) |
| 43 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.988 + 0.149i)T \) |
| 53 | \( 1 + (-0.955 + 0.294i)T \) |
| 59 | \( 1 + (0.0747 + 0.997i)T \) |
| 61 | \( 1 + (-0.955 - 0.294i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.988 + 0.149i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.623 - 0.781i)T \) |
| 89 | \( 1 + (0.365 + 0.930i)T \) |
| 97 | \( 1 - T \) |
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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
−28.8239064880093469805552885741, −26.842618385445702491062711051544, −26.45178485702061253615721429656, −25.09510151915797342800521049728, −24.3399157592110343591877359085, −23.21677218330295991319489579908, −22.3982874539916503271658191477, −21.609455937930736289814611546140, −20.67127001972250805492348691078, −19.347865802684286685416769893574, −18.46672661008066742545580814162, −16.9966549548045382324660220516, −15.999223561627860776606495085260, −14.92710137048431948692944334703, −14.07547624113400251931069980771, −13.266134902914200879047367441339, −11.77761597036597437805223653892, −11.1083348552836617622733587619, −9.85377942894357733599937941867, −8.04201676083750710400139444838, −6.89911273414998469841357770331, −6.01740180242262666601743757262, −4.64707652627465850219320008913, −3.285500888779439637478811709193, −2.276636033827608310415515436410,
1.568720752715189488389311068773, 3.03453215363169031232456568143, 4.65930697042267646077864465538, 5.19861226953295301087675163756, 6.72422103781628555324064256947, 7.94133328262238088923000807678, 9.46595933191351336874117956284, 10.64233834731833482280413691792, 11.959680204830051110962908493495, 12.81913793537627924578979285185, 13.49373979633569898942281310581, 14.995394526510944833175997204086, 15.632580061195583793341586224025, 16.87167570187167894818116048521, 17.86635421058662372366106345303, 19.77454153439614802686784258086, 20.059778885241521936766928753005, 21.23417834096443012432442735950, 22.073842136714816774889600716296, 23.22487926091133145440449589002, 24.01230343504858122517114521847, 24.92966597393937713944771581715, 25.70281727529050107132248292237, 27.26490290416708095847334001720, 28.48741276113409599352591507174