| L(s) = 1 | − 1.35·5-s − 3.73·7-s − 3.46·11-s + 5.65·13-s + 5.24·17-s − 5.91·19-s + 3.74·23-s − 3.15·25-s + 2.15·29-s + 10.5·31-s + 5.07·35-s − 6.12·37-s + 9.96·41-s − 6.73·43-s − 3.69·47-s + 6.95·49-s + 10.3·53-s + 4.70·55-s + 2.38·59-s − 7.77·61-s − 7.67·65-s − 10.1·67-s + 0.525·71-s + 4.96·73-s + 12.9·77-s − 3.47·79-s − 16.2·83-s + ⋯ |
| L(s) = 1 | − 0.607·5-s − 1.41·7-s − 1.04·11-s + 1.56·13-s + 1.27·17-s − 1.35·19-s + 0.780·23-s − 0.630·25-s + 0.399·29-s + 1.89·31-s + 0.857·35-s − 1.00·37-s + 1.55·41-s − 1.02·43-s − 0.538·47-s + 0.993·49-s + 1.42·53-s + 0.634·55-s + 0.310·59-s − 0.996·61-s − 0.952·65-s − 1.24·67-s + 0.0623·71-s + 0.581·73-s + 1.47·77-s − 0.390·79-s − 1.77·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 6.12T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 2.38T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 0.525T + 71T^{2} \) |
| 73 | \( 1 - 4.96T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79037231451894379771897783512, −6.98596900035726732180899733572, −6.21335103627079577900197962345, −5.83246211576037760853461891775, −4.75823218667328801478492280709, −3.87305742042791300550412101169, −3.27493169384653901231368182405, −2.58699909596995142104094880282, −1.12061083300263257502009643979, 0,
1.12061083300263257502009643979, 2.58699909596995142104094880282, 3.27493169384653901231368182405, 3.87305742042791300550412101169, 4.75823218667328801478492280709, 5.83246211576037760853461891775, 6.21335103627079577900197962345, 6.98596900035726732180899733572, 7.79037231451894379771897783512
