| L(s) = 1 | + 2.56·3-s − 0.561·5-s + 3.56·9-s − 2·13-s − 1.43·15-s − 7.12·17-s + 1.12·19-s − 7.68·23-s − 4.68·25-s + 1.43·27-s − 7.12·29-s − 5.43·31-s − 5.68·37-s − 5.12·39-s + 8.24·41-s + 1.12·43-s − 2.00·45-s − 4·47-s − 7·49-s − 18.2·51-s + 8.24·53-s + 2.87·57-s − 0.315·59-s − 9.36·61-s + 1.12·65-s + 7.68·67-s − 19.6·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.251·5-s + 1.18·9-s − 0.554·13-s − 0.371·15-s − 1.72·17-s + 0.257·19-s − 1.60·23-s − 0.936·25-s + 0.276·27-s − 1.32·29-s − 0.976·31-s − 0.934·37-s − 0.820·39-s + 1.28·41-s + 0.171·43-s − 0.298·45-s − 0.583·47-s − 49-s − 2.55·51-s + 1.13·53-s + 0.381·57-s − 0.0410·59-s − 1.19·61-s + 0.139·65-s + 0.938·67-s − 2.36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 0.315T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 0.561T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−8.009393545227707376855643180192, −7.67868811857219355884368623171, −6.86582207649237188466939055514, −5.95881660467502401179607199598, −4.91450632487716122312660707064, −3.93381057263077668863107319882, −3.57488129465475308633917569404, −2.25237098252720304347138178885, −2.01723981897580867765610537240, 0,
2.01723981897580867765610537240, 2.25237098252720304347138178885, 3.57488129465475308633917569404, 3.93381057263077668863107319882, 4.91450632487716122312660707064, 5.95881660467502401179607199598, 6.86582207649237188466939055514, 7.67868811857219355884368623171, 8.009393545227707376855643180192
