| L(s) = 1 | − 1.85·2-s + 3-s + 1.43·4-s − 2.52·5-s − 1.85·6-s + 7-s + 1.04·8-s + 9-s + 4.68·10-s − 0.833·11-s + 1.43·12-s − 4.92·13-s − 1.85·14-s − 2.52·15-s − 4.80·16-s − 5.89·17-s − 1.85·18-s + 7.44·19-s − 3.63·20-s + 21-s + 1.54·22-s − 5.86·23-s + 1.04·24-s + 1.38·25-s + 9.12·26-s + 27-s + 1.43·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.577·3-s + 0.718·4-s − 1.13·5-s − 0.756·6-s + 0.377·7-s + 0.369·8-s + 0.333·9-s + 1.48·10-s − 0.251·11-s + 0.414·12-s − 1.36·13-s − 0.495·14-s − 0.652·15-s − 1.20·16-s − 1.42·17-s − 0.436·18-s + 1.70·19-s − 0.811·20-s + 0.218·21-s + 0.329·22-s − 1.22·23-s + 0.213·24-s + 0.277·25-s + 1.78·26-s + 0.192·27-s + 0.271·28-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 | \( 1 + 1.85T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 + 0.833T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 - 6.55T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 5.78T + 43T^{2} \) |
| 47 | \( 1 + 2.42T + 47T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 0.0392T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 8.05T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 0.933T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.82138584204190617800402889713, −7.20486128133601430604773907852, −6.56463573284458165517403889505, −5.21943658226839692971914777536, −4.51637222560228838136234816748, −3.98069131515292770904284114417, −2.78054315571687931461863788000, −2.17972407290680036685261861556, −0.978951064452393090883803991271, 0,
0.978951064452393090883803991271, 2.17972407290680036685261861556, 2.78054315571687931461863788000, 3.98069131515292770904284114417, 4.51637222560228838136234816748, 5.21943658226839692971914777536, 6.56463573284458165517403889505, 7.20486128133601430604773907852, 7.82138584204190617800402889713
