| L(s) = 1 | + 2-s − 2.36·3-s + 4-s + 1.57·5-s − 2.36·6-s + 1.59·7-s + 8-s + 2.61·9-s + 1.57·10-s − 1.45·11-s − 2.36·12-s + 1.08·13-s + 1.59·14-s − 3.73·15-s + 16-s − 1.26·17-s + 2.61·18-s + 2.02·19-s + 1.57·20-s − 3.77·21-s − 1.45·22-s − 3.02·23-s − 2.36·24-s − 2.52·25-s + 1.08·26-s + 0.908·27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s + 0.703·5-s − 0.967·6-s + 0.602·7-s + 0.353·8-s + 0.872·9-s + 0.497·10-s − 0.438·11-s − 0.684·12-s + 0.302·13-s + 0.425·14-s − 0.963·15-s + 0.250·16-s − 0.306·17-s + 0.616·18-s + 0.464·19-s + 0.351·20-s − 0.824·21-s − 0.310·22-s − 0.630·23-s − 0.483·24-s − 0.504·25-s + 0.213·26-s + 0.174·27-s + 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 - 1.59T + 41T^{2} \) |
| 43 | \( 1 - 0.188T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 + 2.64T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 5.82T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.57T + 73T^{2} \) |
| 79 | \( 1 - 7.53T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 + 0.722T + 89T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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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
−7.37795768411650138139686745412, −6.33281564427501147217893340806, −6.08275427243688107129723344080, −5.35922324928160613525737668703, −4.93527673725258256775918486156, −4.17101441965055780760540249054, −3.19650783116280552769615182010, −2.08918325919378044810264664503, −1.38678227075514912988256521368, 0,
1.38678227075514912988256521368, 2.08918325919378044810264664503, 3.19650783116280552769615182010, 4.17101441965055780760540249054, 4.93527673725258256775918486156, 5.35922324928160613525737668703, 6.08275427243688107129723344080, 6.33281564427501147217893340806, 7.37795768411650138139686745412
