L(s) = 1 | − 3.88·2-s + 9.09·3-s + 7.12·4-s − 19.0·5-s − 35.3·6-s + 18.7·7-s + 3.41·8-s + 55.7·9-s + 74.0·10-s + 4.37·11-s + 64.7·12-s + 45.6·13-s − 73.0·14-s − 173.·15-s − 70.2·16-s − 62.8·17-s − 216.·18-s + 33.1·19-s − 135.·20-s + 170.·21-s − 17.0·22-s − 32.0·23-s + 31.0·24-s + 237.·25-s − 177.·26-s + 261.·27-s + 133.·28-s + ⋯ |
L(s) = 1 | − 1.37·2-s + 1.75·3-s + 0.890·4-s − 1.70·5-s − 2.40·6-s + 1.01·7-s + 0.150·8-s + 2.06·9-s + 2.34·10-s + 0.119·11-s + 1.55·12-s + 0.973·13-s − 1.39·14-s − 2.98·15-s − 1.09·16-s − 0.896·17-s − 2.83·18-s + 0.399·19-s − 1.51·20-s + 1.77·21-s − 0.164·22-s − 0.290·23-s + 0.264·24-s + 1.90·25-s − 1.33·26-s + 1.86·27-s + 0.903·28-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 | \( 1 + 3.88T + 8T^{2} \) |
| 3 | \( 1 - 9.09T + 27T^{2} \) |
| 5 | \( 1 + 19.0T + 125T^{2} \) |
| 7 | \( 1 - 18.7T + 343T^{2} \) |
| 11 | \( 1 - 4.37T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 83.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 90.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 422.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 363.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 248.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 64.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 80.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 936.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 651.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 26.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 347.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 291.T + 9.12e5T^{2} \) |
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
−8.391925891930729295421462301685, −8.080078253221397637040724289759, −7.44769986955650305496626463632, −6.86003433187500958468778203401, −4.87894539206862025277570541870, −3.96101518341460664293157922742, −3.46826231476750935325826906568, −2.09019492116343321049154455126, −1.34100561249662907126907315539, 0,
1.34100561249662907126907315539, 2.09019492116343321049154455126, 3.46826231476750935325826906568, 3.96101518341460664293157922742, 4.87894539206862025277570541870, 6.86003433187500958468778203401, 7.44769986955650305496626463632, 8.080078253221397637040724289759, 8.391925891930729295421462301685