| L(s) = 1 | − 5.01·2-s + 4.57·3-s + 17.1·4-s + 10.5·5-s − 22.9·6-s + 16.1·7-s − 46.0·8-s − 6.03·9-s − 52.8·10-s + 11·11-s + 78.6·12-s − 81.0·14-s + 48.2·15-s + 93.6·16-s + 58.7·17-s + 30.3·18-s − 81.1·19-s + 180.·20-s + 73.9·21-s − 55.1·22-s − 150.·23-s − 210.·24-s − 14.0·25-s − 151.·27-s + 277.·28-s − 188.·29-s − 241.·30-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.881·3-s + 2.14·4-s + 0.941·5-s − 1.56·6-s + 0.871·7-s − 2.03·8-s − 0.223·9-s − 1.67·10-s + 0.301·11-s + 1.89·12-s − 1.54·14-s + 0.829·15-s + 1.46·16-s + 0.837·17-s + 0.396·18-s − 0.979·19-s + 2.02·20-s + 0.768·21-s − 0.534·22-s − 1.36·23-s − 1.79·24-s − 0.112·25-s − 1.07·27-s + 1.87·28-s − 1.20·29-s − 1.47·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.01T + 8T^{2} \) |
| 3 | \( 1 - 4.57T + 27T^{2} \) |
| 5 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 - 16.1T + 343T^{2} \) |
| 17 | \( 1 - 58.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 60.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 11.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 326.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 489.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 94.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 604.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 904.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 499.T + 9.12e5T^{2} \) |
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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
−8.499437489652901776913599442356, −7.986615975701603951327203500186, −7.42974512105673929157923507560, −6.23836159088639854154132915875, −5.67565650184016203470543192134, −4.15482680709005480549032789852, −2.83150421816686271673772815023, −1.97805878246928213590438164827, −1.48645322529039106468963112939, 0,
1.48645322529039106468963112939, 1.97805878246928213590438164827, 2.83150421816686271673772815023, 4.15482680709005480549032789852, 5.67565650184016203470543192134, 6.23836159088639854154132915875, 7.42974512105673929157923507560, 7.986615975701603951327203500186, 8.499437489652901776913599442356
