| L(s) = 1 | − 2-s + 2.40·3-s + 4-s + 1.21·5-s − 2.40·6-s − 4.62·7-s − 8-s + 2.80·9-s − 1.21·10-s + 2.40·12-s + 2.85·13-s + 4.62·14-s + 2.93·15-s + 16-s + 4.99·17-s − 2.80·18-s − 5.76·19-s + 1.21·20-s − 11.1·21-s − 23-s − 2.40·24-s − 3.51·25-s − 2.85·26-s − 0.470·27-s − 4.62·28-s − 2.00·29-s − 2.93·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.39·3-s + 0.5·4-s + 0.545·5-s − 0.983·6-s − 1.74·7-s − 0.353·8-s + 0.934·9-s − 0.385·10-s + 0.695·12-s + 0.792·13-s + 1.23·14-s + 0.758·15-s + 0.250·16-s + 1.21·17-s − 0.661·18-s − 1.32·19-s + 0.272·20-s − 2.43·21-s − 0.208·23-s − 0.491·24-s − 0.702·25-s − 0.560·26-s − 0.0904·27-s − 0.874·28-s − 0.372·29-s − 0.536·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.097347266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097347266\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 29 | \( 1 + 2.00T + 29T^{2} \) |
| 31 | \( 1 - 0.543T + 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 - 9.95T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 2.05T + 71T^{2} \) |
| 73 | \( 1 - 3.16T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 - 7.59T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.298989106980102392469772364400, −7.62911458718204478365205918592, −6.83510223374016377959449255454, −6.14161755495317897742215499775, −5.63021980914396096482709224468, −3.88703126957671369794014697502, −3.62090274930613462525130957827, −2.61824215331389961899200166212, −2.13830100881853969787766116511, −0.78327779748417574490513757837,
0.78327779748417574490513757837, 2.13830100881853969787766116511, 2.61824215331389961899200166212, 3.62090274930613462525130957827, 3.88703126957671369794014697502, 5.63021980914396096482709224468, 6.14161755495317897742215499775, 6.83510223374016377959449255454, 7.62911458718204478365205918592, 8.298989106980102392469772364400
