| L(s) = 1 | − 1.40·2-s + 2.44·3-s − 6.03·4-s − 13.4·5-s − 3.42·6-s − 4.78·7-s + 19.6·8-s − 21.0·9-s + 18.8·10-s − 14.7·12-s − 13·13-s + 6.71·14-s − 32.8·15-s + 20.6·16-s − 44.1·17-s + 29.4·18-s + 124.·19-s + 81.1·20-s − 11.7·21-s − 135.·23-s + 48.1·24-s + 55.9·25-s + 18.2·26-s − 117.·27-s + 28.8·28-s − 75.4·29-s + 46.1·30-s + ⋯ |
| L(s) = 1 | − 0.495·2-s + 0.470·3-s − 0.754·4-s − 1.20·5-s − 0.233·6-s − 0.258·7-s + 0.869·8-s − 0.778·9-s + 0.596·10-s − 0.354·12-s − 0.277·13-s + 0.128·14-s − 0.566·15-s + 0.323·16-s − 0.630·17-s + 0.386·18-s + 1.50·19-s + 0.907·20-s − 0.121·21-s − 1.22·23-s + 0.409·24-s + 0.447·25-s + 0.137·26-s − 0.836·27-s + 0.194·28-s − 0.482·29-s + 0.280·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1866631262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1866631262\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 1.40T + 8T^{2} \) |
| 3 | \( 1 - 2.44T + 27T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 7 | \( 1 + 4.78T + 343T^{2} \) |
| 17 | \( 1 + 44.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 75.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 393.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 160.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 191.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 81.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 61.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 167.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 936.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 103.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 938.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 286.T + 9.12e5T^{2} \) |
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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
−8.991484820010026043101082773579, −8.269358960082664812068493757349, −7.74587261266047576663204428369, −7.02457305194999760314327234921, −5.66084759739100618218852283565, −4.83769620270317408527862865587, −3.75692810614147336999433546408, −3.29966705835162609426421910993, −1.78984957490119028233512596749, −0.21579967869048140053432577078,
0.21579967869048140053432577078, 1.78984957490119028233512596749, 3.29966705835162609426421910993, 3.75692810614147336999433546408, 4.83769620270317408527862865587, 5.66084759739100618218852283565, 7.02457305194999760314327234921, 7.74587261266047576663204428369, 8.269358960082664812068493757349, 8.991484820010026043101082773579
