| 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 | \( 1 + (65.8 + 23.9i)T + (6.99e5 + 5.86e5i)T^{2} \) |
| 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
−12.56450579977227497077669408978, −10.55588191532761614394206356038, −10.20543222698166054546572025670, −9.250831513263268963476716365626, −8.118481323009833615757602564472, −6.88014089723220426094656583076, −5.77462804053012618829561972662, −4.80813195897159991295118834232, −2.41665626340597243058985517622, −0.76695734501635442012047189093,
2.03887910921043483374434308449, 2.68911154609997244414289033074, 5.05988109384823670062812113097, 6.21344296161356290312554859454, 7.27826486676892228427327894940, 8.970099187541053370963721883905, 9.593041722404423127673638300175, 10.30275326266192459154515371140, 11.50518342567888861164628687678, 12.52364625614076196637011215684
