| L(s) = 1 | + 2i·7-s − 3·9-s − 5.29i·11-s − 2i·13-s + 5.29i·19-s + 6i·23-s + (1 + 5.29i)29-s − 5.29i·31-s + 10.5·43-s + 10.5·47-s + 3·49-s − 2i·53-s + 10.5i·61-s − 6i·63-s − 10i·67-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 9-s − 1.59i·11-s − 0.554i·13-s + 1.21i·19-s + 1.25i·23-s + (0.185 + 0.982i)29-s − 0.950i·31-s + 1.61·43-s + 1.54·47-s + 0.428·49-s − 0.274i·53-s + 1.35i·61-s − 0.755i·63-s − 1.22i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.478631902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478631902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (-1 - 5.29i)T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 31 | \( 1 + 5.29iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.896004967194897999980055271047, −8.117956959825953543512511704393, −7.58568368671509020840919385452, −6.26329988492438165345672620128, −5.67106317916634771298399570750, −5.41168496311763813989856989404, −3.88038921954267780369696330675, −3.18868715331806131543695533947, −2.36402714696893566000571701884, −0.900403117926518157542476321706,
0.60566064161012272725238326515, 2.10539806589814186597362563549, 2.84687584621060324411633649944, 4.21200406777073394533711212557, 4.56783277671547961502270804820, 5.59545759271579901102910687852, 6.60816154404064747166811610169, 7.11192102409330456159793662312, 7.84973554402520100105944893700, 8.814659765288793826620538323127
