| L(s) = 1 | − 0.611·3-s − 1.62·5-s + 3.10·7-s − 2.62·9-s + 5.31·11-s + 13-s + 0.994·15-s − 1.89·17-s + 0.885·19-s − 1.89·21-s − 1.22·23-s − 2.35·25-s + 3.43·27-s + 2·29-s − 3.71·31-s − 3.25·33-s − 5.04·35-s + 11.1·37-s − 0.611·39-s + 5.79·41-s + 3.43·43-s + 4.27·45-s + 0.274·47-s + 2.62·49-s + 1.15·51-s + 7.79·53-s − 8.65·55-s + ⋯ |
| L(s) = 1 | − 0.352·3-s − 0.727·5-s + 1.17·7-s − 0.875·9-s + 1.60·11-s + 0.277·13-s + 0.256·15-s − 0.460·17-s + 0.203·19-s − 0.413·21-s − 0.254·23-s − 0.471·25-s + 0.661·27-s + 0.371·29-s − 0.667·31-s − 0.566·33-s − 0.852·35-s + 1.83·37-s − 0.0979·39-s + 0.904·41-s + 0.524·43-s + 0.636·45-s + 0.0399·47-s + 0.375·49-s + 0.162·51-s + 1.07·53-s − 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.625040731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625040731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.611T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 0.885T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 - 0.274T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 7.04T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 8.08T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 5.25T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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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
−8.760569505679042971861330195073, −7.82335876231952971589944682021, −7.30678707207996203153067878632, −6.18901541960992470942086202594, −5.78007148528896867705819570788, −4.53686904309086578594722414429, −4.20565265490568762704411645251, −3.13393754920328011202412473763, −1.89161096189450299477721075369, −0.800410563065936705852700341572,
0.800410563065936705852700341572, 1.89161096189450299477721075369, 3.13393754920328011202412473763, 4.20565265490568762704411645251, 4.53686904309086578594722414429, 5.78007148528896867705819570788, 6.18901541960992470942086202594, 7.30678707207996203153067878632, 7.82335876231952971589944682021, 8.760569505679042971861330195073
