L(s) = 1 | + (2.77 − 4.15i)5-s + (−8.02 + 4.63i)7-s + (13.2 − 7.64i)11-s + (−2.59 − 1.5i)13-s − 4.00·17-s − 27.7·19-s + (−18.9 + 32.7i)23-s + (−9.56 − 23.0i)25-s + (39.7 − 22.9i)29-s + (−14.3 + 24.9i)31-s + (−3.02 + 46.2i)35-s + 34.3i·37-s + (−23.6 − 13.6i)41-s + (31.4 − 18.1i)43-s + (10.0 + 17.3i)47-s + ⋯ |
L(s) = 1 | + (0.555 − 0.831i)5-s + (−1.14 + 0.661i)7-s + (1.20 − 0.695i)11-s + (−0.199 − 0.115i)13-s − 0.235·17-s − 1.46·19-s + (−0.823 + 1.42i)23-s + (−0.382 − 0.923i)25-s + (1.37 − 0.791i)29-s + (−0.464 + 0.804i)31-s + (−0.0865 + 1.32i)35-s + 0.927i·37-s + (−0.577 − 0.333i)41-s + (0.730 − 0.421i)43-s + (0.213 + 0.369i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5305355227\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5305355227\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.77 + 4.15i)T \) |
good | 7 | \( 1 + (8.02 - 4.63i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-13.2 + 7.64i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (2.59 + 1.5i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 4.00T + 289T^{2} \) |
| 19 | \( 1 + 27.7T + 361T^{2} \) |
| 23 | \( 1 + (18.9 - 32.7i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-39.7 + 22.9i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (14.3 - 24.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 34.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (23.6 + 13.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-31.4 + 18.1i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-10.0 - 17.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 57.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (73.8 + 42.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-18.8 - 32.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-50.4 - 29.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (19.5 + 33.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-32.9 - 57.0i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 85.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (140. - 80.9i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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
−9.491689069454432922847318037726, −8.740695522696822312053994575750, −8.232656809990409412983281684650, −6.78326723949703687087037578568, −6.20649789396084728385920427122, −5.61319952785403758753329228844, −4.46056719739428049231263645970, −3.56416298631038320652211216394, −2.44279568503475959712329920142, −1.28030449292812017778275572926,
0.13982740637309627018937859861, 1.80831843321087934300750936833, 2.78496618200954976091996506998, 3.86790603965797812985804719812, 4.52202147018222200571438246055, 6.06370874440695118362064782416, 6.60833108533156984579499739727, 6.93617337050423259280198816484, 8.112371111262731357427154681044, 9.227167431271669220840768193216