| L(s) = 1 | − 2.03·2-s + 3-s + 2.12·4-s − 1.52·5-s − 2.03·6-s − 7-s − 0.259·8-s + 9-s + 3.10·10-s − 4.20·11-s + 2.12·12-s + 5.16·13-s + 2.03·14-s − 1.52·15-s − 3.72·16-s − 3.56·17-s − 2.03·18-s − 0.432·19-s − 3.25·20-s − 21-s + 8.55·22-s − 0.283·23-s − 0.259·24-s − 2.66·25-s − 10.5·26-s + 27-s − 2.12·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 0.577·3-s + 1.06·4-s − 0.683·5-s − 0.829·6-s − 0.377·7-s − 0.0917·8-s + 0.333·9-s + 0.982·10-s − 1.26·11-s + 0.614·12-s + 1.43·13-s + 0.542·14-s − 0.394·15-s − 0.932·16-s − 0.863·17-s − 0.478·18-s − 0.0993·19-s − 0.727·20-s − 0.218·21-s + 1.82·22-s − 0.0590·23-s − 0.0529·24-s − 0.532·25-s − 2.05·26-s + 0.192·27-s − 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 0.432T + 19T^{2} \) |
| 23 | \( 1 + 0.283T + 23T^{2} \) |
| 29 | \( 1 + 0.577T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 - 5.92T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 + 8.16T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.83624832693204585704989311573, −7.09863771903311593295841954707, −6.43772356585945024207487943958, −5.54977135802863519265823493819, −4.42097207167884616865802500029, −3.84719392030881871784894850727, −2.83111342414423277214513292196, −2.12186863297121092217768354374, −1.00629833870301078988314620292, 0,
1.00629833870301078988314620292, 2.12186863297121092217768354374, 2.83111342414423277214513292196, 3.84719392030881871784894850727, 4.42097207167884616865802500029, 5.54977135802863519265823493819, 6.43772356585945024207487943958, 7.09863771903311593295841954707, 7.83624832693204585704989311573
