| L(s) = 1 | + 0.879·2-s − 3-s − 1.22·4-s + 1.34·5-s − 0.879·6-s − 1.22·7-s − 2.83·8-s + 9-s + 1.18·10-s + 4.29·11-s + 1.22·12-s − 5.34·13-s − 1.07·14-s − 1.34·15-s − 0.0418·16-s + 0.879·18-s − 5.59·19-s − 1.65·20-s + 1.22·21-s + 3.77·22-s − 5.94·23-s + 2.83·24-s − 3.18·25-s − 4.70·26-s − 27-s + 1.50·28-s + 4.02·29-s + ⋯ |
| L(s) = 1 | + 0.621·2-s − 0.577·3-s − 0.613·4-s + 0.602·5-s − 0.359·6-s − 0.463·7-s − 1.00·8-s + 0.333·9-s + 0.374·10-s + 1.29·11-s + 0.354·12-s − 1.48·13-s − 0.288·14-s − 0.347·15-s − 0.0104·16-s + 0.207·18-s − 1.28·19-s − 0.369·20-s + 0.267·21-s + 0.804·22-s − 1.23·23-s + 0.579·24-s − 0.636·25-s − 0.922·26-s − 0.192·27-s + 0.284·28-s + 0.746·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.879T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 + 7.30T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 + 9.87T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 - 9.47T + 61T^{2} \) |
| 67 | \( 1 - 0.128T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 3.81T + 73T^{2} \) |
| 79 | \( 1 + 1.36T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−9.825645038054744066965552062693, −9.111472799426605538510859886259, −8.108913837563353263149397631989, −6.64060516786305159409560154193, −6.29785521339682511941642059890, −5.20737777985455940890191870646, −4.45670900800904651242038306179, −3.49920385972607914206303306420, −1.99484173185798198962115494581, 0,
1.99484173185798198962115494581, 3.49920385972607914206303306420, 4.45670900800904651242038306179, 5.20737777985455940890191870646, 6.29785521339682511941642059890, 6.64060516786305159409560154193, 8.108913837563353263149397631989, 9.111472799426605538510859886259, 9.825645038054744066965552062693
