L(s) = 1 | + (−1.53 − 1.28i)2-s + (0.694 + 3.93i)4-s + (19.8 + 7.22i)5-s + (−3.38 + 19.1i)7-s + (4.00 − 6.92i)8-s + (−21.1 − 36.5i)10-s + (−20.6 + 7.51i)11-s + (−57.4 + 48.2i)13-s + (29.8 − 25.0i)14-s + (−15.0 + 5.47i)16-s + (−22.5 − 39.1i)17-s + (−10.9 + 18.9i)19-s + (−14.6 + 83.1i)20-s + (41.3 + 15.0i)22-s + (9.15 + 51.9i)23-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (1.77 + 0.645i)5-s + (−0.182 + 1.03i)7-s + (0.176 − 0.306i)8-s + (−0.667 − 1.15i)10-s + (−0.566 + 0.206i)11-s + (−1.22 + 1.02i)13-s + (0.569 − 0.478i)14-s + (−0.234 + 0.0855i)16-s + (−0.322 − 0.558i)17-s + (−0.131 + 0.228i)19-s + (−0.163 + 0.929i)20-s + (0.400 + 0.145i)22-s + (0.0829 + 0.470i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.17468 + 0.780211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17468 + 0.780211i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.53 + 1.28i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-19.8 - 7.22i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (3.38 - 19.1i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (20.6 - 7.51i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (57.4 - 48.2i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (22.5 + 39.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (10.9 - 18.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-9.15 - 51.9i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-92.8 - 77.8i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (26.8 + 152. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-91.7 - 158. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-213. + 178. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (188. - 68.5i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (3.39 - 19.2i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 646.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (415. + 151. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (21.3 - 121. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-222. + 186. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-40.8 - 70.6i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-170. + 295. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-907. - 761. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (716. + 601. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-704. + 1.21e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (65.8 - 23.9i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.52364625614076196637011215684, −11.50518342567888861164628687678, −10.30275326266192459154515371140, −9.593041722404423127673638300175, −8.970099187541053370963721883905, −7.27826486676892228427327894940, −6.21344296161356290312554859454, −5.05988109384823670062812113097, −2.68911154609997244414289033074, −2.03887910921043483374434308449,
0.76695734501635442012047189093, 2.41665626340597243058985517622, 4.80813195897159991295118834232, 5.77462804053012618829561972662, 6.88014089723220426094656583076, 8.118481323009833615757602564472, 9.250831513263268963476716365626, 10.20543222698166054546572025670, 10.55588191532761614394206356038, 12.56450579977227497077669408978