L(s) = 1 | + (0.144 − 0.412i)2-s + (2.97 + 2.37i)4-s + (1.02 + 2.13i)5-s + (−1.74 − 2.18i)7-s + (2.89 − 1.81i)8-s + (1.02 − 0.115i)10-s + (7.71 + 4.84i)11-s + (8.82 − 2.01i)13-s + (−1.15 + 0.404i)14-s + (3.05 + 13.3i)16-s + (−5.09 + 5.09i)17-s + (2.57 + 22.8i)19-s + (−2.00 + 8.79i)20-s + (3.11 − 2.48i)22-s + (−17.8 − 8.59i)23-s + ⋯ |
L(s) = 1 | + (0.0722 − 0.206i)2-s + (0.744 + 0.593i)4-s + (0.205 + 0.426i)5-s + (−0.249 − 0.312i)7-s + (0.361 − 0.227i)8-s + (0.102 − 0.0115i)10-s + (0.701 + 0.440i)11-s + (0.679 − 0.155i)13-s + (−0.0825 + 0.0288i)14-s + (0.191 + 0.837i)16-s + (−0.299 + 0.299i)17-s + (0.135 + 1.20i)19-s + (−0.100 + 0.439i)20-s + (0.141 − 0.112i)22-s + (−0.776 − 0.373i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.99190 + 0.575156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99190 + 0.575156i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 + (-26.8 - 10.9i)T \) |
good | 2 | \( 1 + (-0.144 + 0.412i)T + (-3.12 - 2.49i)T^{2} \) |
| 5 | \( 1 + (-1.02 - 2.13i)T + (-15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (1.74 + 2.18i)T + (-10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-7.71 - 4.84i)T + (52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (-8.82 + 2.01i)T + (152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (5.09 - 5.09i)T - 289iT^{2} \) |
| 19 | \( 1 + (-2.57 - 22.8i)T + (-351. + 80.3i)T^{2} \) |
| 23 | \( 1 + (17.8 + 8.59i)T + (329. + 413. i)T^{2} \) |
| 31 | \( 1 + (10.2 - 29.3i)T + (-751. - 599. i)T^{2} \) |
| 37 | \( 1 + (-5.78 + 3.63i)T + (593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (9.53 + 9.53i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-47.8 + 16.7i)T + (1.44e3 - 1.15e3i)T^{2} \) |
| 47 | \( 1 + (-44.9 + 71.5i)T + (-958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (76.8 - 36.9i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 + 54.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + (85.8 + 9.67i)T + (3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (70.4 + 16.0i)T + (4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-85.2 + 19.4i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (41.8 + 119. i)T + (-4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (29.8 + 47.4i)T + (-2.70e3 + 5.62e3i)T^{2} \) |
| 83 | \( 1 + (57.5 - 72.2i)T + (-1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-31.8 + 91.0i)T + (-6.19e3 - 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-17.8 + 2.00i)T + (9.17e3 - 2.09e3i)T^{2} \) |
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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
−12.07189187049404834793470570207, −10.70463584848531335076280633433, −10.35342881982525135882620043167, −8.902977520112630416800873612375, −7.84291916236330581484633704090, −6.79287278411843065433068476999, −6.09320842229428653564574748325, −4.20839158488503976071341880437, −3.20332859910540920301065357206, −1.73642345723823389981966632297,
1.20162062893095168053128557816, 2.77614745351037598587553482883, 4.49865486337006217141597711232, 5.80531290150465405543982446421, 6.45234582371502668514175983154, 7.62233534905081940993948011523, 8.956641639111785765954592519982, 9.614935051792715004685174378313, 10.96352638920861439754943832676, 11.46914283272984279650845749146