| L(s) = 1 | − 1.29·2-s + 3.36·3-s − 6.32·4-s − 1.78·5-s − 4.34·6-s + 27.5·7-s + 18.5·8-s − 15.7·9-s + 2.30·10-s + 9.45·11-s − 21.2·12-s + 38.4·13-s − 35.6·14-s − 5.98·15-s + 26.6·16-s + 54.0·17-s + 20.3·18-s − 37.3·19-s + 11.2·20-s + 92.6·21-s − 12.2·22-s − 100.·23-s + 62.2·24-s − 121.·25-s − 49.7·26-s − 143.·27-s − 174.·28-s + ⋯ |
| L(s) = 1 | − 0.457·2-s + 0.646·3-s − 0.790·4-s − 0.159·5-s − 0.295·6-s + 1.48·7-s + 0.819·8-s − 0.581·9-s + 0.0729·10-s + 0.259·11-s − 0.511·12-s + 0.820·13-s − 0.681·14-s − 0.103·15-s + 0.415·16-s + 0.771·17-s + 0.266·18-s − 0.450·19-s + 0.125·20-s + 0.962·21-s − 0.118·22-s − 0.915·23-s + 0.529·24-s − 0.974·25-s − 0.375·26-s − 1.02·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.29T + 8T^{2} \) |
| 3 | \( 1 - 3.36T + 27T^{2} \) |
| 5 | \( 1 + 1.78T + 125T^{2} \) |
| 7 | \( 1 - 27.5T + 343T^{2} \) |
| 11 | \( 1 - 9.45T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 234.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 325.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 148.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 735.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 275.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 30.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 27.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 66.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 511.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 144.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 769.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.88e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.636861171857657923251768522066, −7.81933798493606012495555748274, −7.49670066365520032625238762234, −5.82379249681059228776490648267, −5.32871765781848081978307428239, −4.08321938994375114983241962901, −3.70780888223751038177020753096, −2.13027541491782549978014478057, −1.34245819469046152027103600893, 0,
1.34245819469046152027103600893, 2.13027541491782549978014478057, 3.70780888223751038177020753096, 4.08321938994375114983241962901, 5.32871765781848081978307428239, 5.82379249681059228776490648267, 7.49670066365520032625238762234, 7.81933798493606012495555748274, 8.636861171857657923251768522066
