L(s) = 1 | − 2.14·2-s + 23.1·3-s − 27.4·4-s − 49.6·6-s − 101.·7-s + 127.·8-s + 294.·9-s + 303.·11-s − 635.·12-s + 946.·13-s + 216.·14-s + 604.·16-s − 661.·17-s − 629.·18-s + 1.28e3·19-s − 2.34e3·21-s − 649.·22-s − 963.·23-s + 2.94e3·24-s − 2.02e3·26-s + 1.18e3·27-s + 2.77e3·28-s + 2.73e3·29-s + 2.49e3·31-s − 5.36e3·32-s + 7.02e3·33-s + 1.41e3·34-s + ⋯ |
L(s) = 1 | − 0.378·2-s + 1.48·3-s − 0.856·4-s − 0.562·6-s − 0.781·7-s + 0.702·8-s + 1.20·9-s + 0.755·11-s − 1.27·12-s + 1.55·13-s + 0.295·14-s + 0.590·16-s − 0.555·17-s − 0.457·18-s + 0.818·19-s − 1.16·21-s − 0.285·22-s − 0.379·23-s + 1.04·24-s − 0.588·26-s + 0.312·27-s + 0.669·28-s + 0.604·29-s + 0.466·31-s − 0.926·32-s + 1.12·33-s + 0.210·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.828307712\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.828307712\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
good | 2 | \( 1 + 2.14T + 32T^{2} \) |
| 3 | \( 1 - 23.1T + 243T^{2} \) |
| 7 | \( 1 + 101.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 303.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 946.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 661.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 963.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.49e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.22e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 52.2T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.48e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.88e4T + 8.58e9T^{2} \) |
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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
−9.047007045852230811190154785514, −8.544848328314010982330384409744, −7.87184100783840715049652681612, −6.82862158895918765074987036319, −5.88515702169416887982972872501, −4.46008668368305926534523163730, −3.69078192096604034688709456084, −3.08663134728049984748824603404, −1.71730226726620372992842150650, −0.75066789420808810219582511976,
0.75066789420808810219582511976, 1.71730226726620372992842150650, 3.08663134728049984748824603404, 3.69078192096604034688709456084, 4.46008668368305926534523163730, 5.88515702169416887982972872501, 6.82862158895918765074987036319, 7.87184100783840715049652681612, 8.544848328314010982330384409744, 9.047007045852230811190154785514