| L(s) = 1 | − 2.74·2-s + 3-s + 5.51·4-s − 1.10·5-s − 2.74·6-s + 7-s − 9.63·8-s + 9-s + 3.01·10-s − 4.21·11-s + 5.51·12-s − 3.35·13-s − 2.74·14-s − 1.10·15-s + 15.3·16-s + 2.96·17-s − 2.74·18-s + 7.68·19-s − 6.06·20-s + 21-s + 11.5·22-s − 1.79·23-s − 9.63·24-s − 3.78·25-s + 9.19·26-s + 27-s + 5.51·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.75·4-s − 0.492·5-s − 1.11·6-s + 0.377·7-s − 3.40·8-s + 0.333·9-s + 0.954·10-s − 1.27·11-s + 1.59·12-s − 0.930·13-s − 0.732·14-s − 0.284·15-s + 3.84·16-s + 0.718·17-s − 0.646·18-s + 1.76·19-s − 1.35·20-s + 0.218·21-s + 2.46·22-s − 0.373·23-s − 1.96·24-s − 0.757·25-s + 1.80·26-s + 0.192·27-s + 1.04·28-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 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 13 | \( 1 + 3.35T + 13T^{2} \) |
| 17 | \( 1 - 2.96T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 + 0.450T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 6.62T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 9.43T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.75294419993085693881343613101, −7.27649151630460290193188952006, −6.56420172761932693220710778165, −5.50559908537088767177598524794, −4.81636666062399208685701426715, −3.31136719987399849234050033896, −2.88632268492725574050658969700, −2.01477321345325071353648083058, −1.08830498790434409172967646636, 0,
1.08830498790434409172967646636, 2.01477321345325071353648083058, 2.88632268492725574050658969700, 3.31136719987399849234050033896, 4.81636666062399208685701426715, 5.50559908537088767177598524794, 6.56420172761932693220710778165, 7.27649151630460290193188952006, 7.75294419993085693881343613101
