| L(s) = 1 | − 1.46·2-s − 1.50·3-s + 0.157·4-s − 5-s + 2.21·6-s − 2.37·7-s + 2.70·8-s − 0.731·9-s + 1.46·10-s + 11-s − 0.237·12-s + 1.65·13-s + 3.48·14-s + 1.50·15-s − 4.29·16-s + 4.24·17-s + 1.07·18-s + 19-s − 0.157·20-s + 3.57·21-s − 1.46·22-s + 5.50·23-s − 4.07·24-s + 25-s − 2.43·26-s + 5.62·27-s − 0.373·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.869·3-s + 0.0787·4-s − 0.447·5-s + 0.903·6-s − 0.895·7-s + 0.956·8-s − 0.243·9-s + 0.464·10-s + 0.301·11-s − 0.0685·12-s + 0.460·13-s + 0.930·14-s + 0.388·15-s − 1.07·16-s + 1.02·17-s + 0.253·18-s + 0.229·19-s − 0.0352·20-s + 0.779·21-s − 0.313·22-s + 1.14·23-s − 0.832·24-s + 0.200·25-s − 0.478·26-s + 1.08·27-s − 0.0705·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 + 8.58T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 3.87T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + 1.67T + 97T^{2} \) |
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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
−9.341977249623408925400389173485, −8.979684853847248585312178858731, −7.79814128427774868452642760036, −7.21114293162802847386730373319, −6.13127765452449601156980843533, −5.37891850644950716376771549788, −4.17930482667588661401070298263, −3.10120834920248443611495545651, −1.19171860538897375701992711888, 0,
1.19171860538897375701992711888, 3.10120834920248443611495545651, 4.17930482667588661401070298263, 5.37891850644950716376771549788, 6.13127765452449601156980843533, 7.21114293162802847386730373319, 7.79814128427774868452642760036, 8.979684853847248585312178858731, 9.341977249623408925400389173485
