| L(s) = 1 | − 0.381·2-s − 1.85·4-s − 1.61·5-s + 7-s + 1.47·8-s + 0.618·10-s + 4.23·13-s − 0.381·14-s + 3.14·16-s − 7.85·17-s + 0.854·19-s + 3·20-s + 4.23·23-s − 2.38·25-s − 1.61·26-s − 1.85·28-s − 6·29-s + 5.09·31-s − 4.14·32-s + 3·34-s − 1.61·35-s − 1.76·37-s − 0.326·38-s − 2.38·40-s − 4.23·41-s − 6.70·43-s − 1.61·46-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.927·4-s − 0.723·5-s + 0.377·7-s + 0.520·8-s + 0.195·10-s + 1.17·13-s − 0.102·14-s + 0.786·16-s − 1.90·17-s + 0.195·19-s + 0.670·20-s + 0.883·23-s − 0.476·25-s − 0.317·26-s − 0.350·28-s − 1.11·29-s + 0.914·31-s − 0.732·32-s + 0.514·34-s − 0.273·35-s − 0.289·37-s − 0.0529·38-s − 0.376·40-s − 0.661·41-s − 1.02·43-s − 0.238·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 + 8.56T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−9.206198600866546643197551946803, −8.660450202551025823236561220985, −8.038542819118388514074790261373, −7.10043806222392998145215181884, −6.06522029000236105675578651626, −4.88174600057505279677996520638, −4.23981348322015757304699491696, −3.31170777820191558925229710888, −1.57145352940805512369100782903, 0,
1.57145352940805512369100782903, 3.31170777820191558925229710888, 4.23981348322015757304699491696, 4.88174600057505279677996520638, 6.06522029000236105675578651626, 7.10043806222392998145215181884, 8.038542819118388514074790261373, 8.660450202551025823236561220985, 9.206198600866546643197551946803
