| L(s) = 1 | + 1.21·2-s + 3-s − 0.534·4-s + 3.03·5-s + 1.21·6-s − 3.40·7-s − 3.06·8-s + 9-s + 3.67·10-s − 3.85·11-s − 0.534·12-s − 5.15·13-s − 4.12·14-s + 3.03·15-s − 2.64·16-s − 2.07·17-s + 1.21·18-s − 0.0953·19-s − 1.62·20-s − 3.40·21-s − 4.66·22-s + 23-s − 3.06·24-s + 4.22·25-s − 6.24·26-s + 27-s + 1.82·28-s + ⋯ |
| L(s) = 1 | + 0.856·2-s + 0.577·3-s − 0.267·4-s + 1.35·5-s + 0.494·6-s − 1.28·7-s − 1.08·8-s + 0.333·9-s + 1.16·10-s − 1.16·11-s − 0.154·12-s − 1.43·13-s − 1.10·14-s + 0.784·15-s − 0.661·16-s − 0.503·17-s + 0.285·18-s − 0.0218·19-s − 0.362·20-s − 0.743·21-s − 0.995·22-s + 0.208·23-s − 0.626·24-s + 0.845·25-s − 1.22·26-s + 0.192·27-s + 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 0.0953T + 19T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.67T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 2.25T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 0.679T + 79T^{2} \) |
| 83 | \( 1 - 4.99T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−9.148072033930318349995899269289, −7.987431199710081814329568644024, −6.95531066663036131675161076552, −6.26375253028128632612957427214, −5.37766231659673463634664555311, −4.89581595556352822188163168416, −3.66120216551366245954769805317, −2.78520282663666424411302311339, −2.21442877862716523103219016889, 0,
2.21442877862716523103219016889, 2.78520282663666424411302311339, 3.66120216551366245954769805317, 4.89581595556352822188163168416, 5.37766231659673463634664555311, 6.26375253028128632612957427214, 6.95531066663036131675161076552, 7.987431199710081814329568644024, 9.148072033930318349995899269289
