| L(s) = 1 | + (−0.347 + 1.96i)2-s + (−3.75 − 1.36i)4-s + (−5.41 + 4.54i)5-s + (−4.71 + 1.71i)7-s + (4 − 6.92i)8-s + (−7.06 − 12.2i)10-s + (−12.3 − 10.3i)11-s + (−3.15 − 17.8i)13-s + (−1.74 − 9.88i)14-s + (12.2 + 10.2i)16-s + (−42.7 − 74.0i)17-s + (55.6 − 96.3i)19-s + (26.5 − 9.66i)20-s + (24.6 − 20.6i)22-s + (−168. − 61.2i)23-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.469 − 0.171i)4-s + (−0.484 + 0.406i)5-s + (−0.254 + 0.0927i)7-s + (0.176 − 0.306i)8-s + (−0.223 − 0.387i)10-s + (−0.338 − 0.283i)11-s + (−0.0672 − 0.381i)13-s + (−0.0332 − 0.188i)14-s + (0.191 + 0.160i)16-s + (−0.609 − 1.05i)17-s + (0.671 − 1.16i)19-s + (0.296 − 0.108i)20-s + (0.239 − 0.200i)22-s + (−1.52 − 0.555i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0427 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.334854 - 0.320824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.334854 - 0.320824i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 - 1.96i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (5.41 - 4.54i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (4.71 - 1.71i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (12.3 + 10.3i)T + (231. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (3.15 + 17.8i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (42.7 + 74.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-55.6 + 96.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (168. + 61.2i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-27.8 + 157. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-116. - 42.4i)T + (2.28e4 + 1.91e4i)T^{2} \) |
| 37 | \( 1 + (120. + 208. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-28.3 - 160. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (150. + 126. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-242. + 88.2i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 - 170.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (448. - 376. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (745. - 271. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-61.1 - 346. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (474. + 821. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (133. - 231. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (191. - 1.08e3i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (165. - 938. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-299. + 519. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.39e3 + 1.17e3i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−12.13202425250383937089637851353, −11.15221136027413151960034012882, −10.00864721434002788812582765678, −8.962603791883187248468709663035, −7.81727016210654742729858045899, −6.96687184774828596995487334761, −5.76785080442446191472768846305, −4.45096520734765257894779660592, −2.86756114077094033662166477805, −0.22354192733959685243496654495,
1.73385195238456923560168065611, 3.52896304073033885787502943038, 4.61781987287747852466724831806, 6.14426438114699890061568678945, 7.72386577966971308213455967531, 8.563700206141702805132019851698, 9.805614396698530171385699643848, 10.56175264081456484789830929688, 11.86798307405093455243135314980, 12.35589369836056577817922886552
