| L(s) = 1 | − 2.63·2-s − 2.63·3-s + 4.94·4-s + 1.42·5-s + 6.94·6-s − 7.77·8-s + 3.94·9-s − 3.76·10-s + 0.693·11-s − 13.0·12-s + 4.78·13-s − 3.76·15-s + 10.5·16-s − 5.68·17-s − 10.4·18-s + 7.99·19-s + 7.07·20-s − 1.82·22-s − 1.82·23-s + 20.4·24-s − 2.95·25-s − 12.6·26-s − 2.49·27-s + 6.53·29-s + 9.93·30-s + 6.82·31-s − 12.3·32-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 1.52·3-s + 2.47·4-s + 0.639·5-s + 2.83·6-s − 2.74·8-s + 1.31·9-s − 1.19·10-s + 0.209·11-s − 3.76·12-s + 1.32·13-s − 0.972·15-s + 2.64·16-s − 1.37·17-s − 2.45·18-s + 1.83·19-s + 1.58·20-s − 0.389·22-s − 0.380·23-s + 4.18·24-s − 0.591·25-s − 2.47·26-s − 0.480·27-s + 1.21·29-s + 1.81·30-s + 1.22·31-s − 2.19·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5136057626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5136057626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 2.63T + 3T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 11 | \( 1 - 0.693T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 7.99T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 43 | \( 1 - 9.73T + 43T^{2} \) |
| 47 | \( 1 - 9.41T + 47T^{2} \) |
| 53 | \( 1 - 0.481T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 + 0.300T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 3.57T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−9.161853449420551172301857627021, −8.625165156990595666162184171473, −7.56874199735644822834950713496, −6.84214078864103837529410288059, −6.07164097116837066197934091957, −5.78487208617982504916266880815, −4.40269570068255564644645599631, −2.78830081667923031381915902329, −1.50024235156982695324664027191, −0.74115785585808448543716389865,
0.74115785585808448543716389865, 1.50024235156982695324664027191, 2.78830081667923031381915902329, 4.40269570068255564644645599631, 5.78487208617982504916266880815, 6.07164097116837066197934091957, 6.84214078864103837529410288059, 7.56874199735644822834950713496, 8.625165156990595666162184171473, 9.161853449420551172301857627021
