| L(s) = 1 | − 3.63·2-s + 11.8·3-s − 18.7·4-s + 25·5-s − 43.1·6-s − 246.·7-s + 184.·8-s − 102.·9-s − 90.9·10-s + 121·11-s − 222.·12-s + 388.·13-s + 897.·14-s + 296.·15-s − 72.1·16-s − 713.·17-s + 373.·18-s − 361·19-s − 468.·20-s − 2.92e3·21-s − 440.·22-s + 1.72e3·23-s + 2.18e3·24-s + 625·25-s − 1.41e3·26-s − 4.09e3·27-s + 4.62e3·28-s + ⋯ |
| L(s) = 1 | − 0.643·2-s + 0.759·3-s − 0.586·4-s + 0.447·5-s − 0.488·6-s − 1.90·7-s + 1.02·8-s − 0.422·9-s − 0.287·10-s + 0.301·11-s − 0.445·12-s + 0.637·13-s + 1.22·14-s + 0.339·15-s − 0.0704·16-s − 0.598·17-s + 0.271·18-s − 0.229·19-s − 0.262·20-s − 1.44·21-s − 0.193·22-s + 0.678·23-s + 0.775·24-s + 0.200·25-s − 0.410·26-s − 1.08·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 3.63T + 32T^{2} \) |
| 3 | \( 1 - 11.8T + 243T^{2} \) |
| 7 | \( 1 + 246.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 388.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 713.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.72e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.16e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.60e4T + 8.58e9T^{2} \) |
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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
−8.738763529956462436613626736894, −8.484753773353921211866164793523, −7.07764010471640799606516941959, −6.44490493891854872817863883716, −5.44558231447916357858966783966, −4.11840875487891105372192866661, −3.29195821607236455018461522182, −2.44021877140407468369229640696, −0.998214781189289942667061919763, 0,
0.998214781189289942667061919763, 2.44021877140407468369229640696, 3.29195821607236455018461522182, 4.11840875487891105372192866661, 5.44558231447916357858966783966, 6.44490493891854872817863883716, 7.07764010471640799606516941959, 8.484753773353921211866164793523, 8.738763529956462436613626736894
