| L(s) = 1 | + (0.347 + 1.96i)2-s + (−3.75 + 1.36i)4-s + (−7.73 − 6.49i)5-s + (7.54 + 2.74i)7-s + (−4 − 6.92i)8-s + (10.1 − 17.4i)10-s + (−12.9 + 10.8i)11-s + (14.9 − 84.8i)13-s + (−2.79 + 15.8i)14-s + (12.2 − 10.2i)16-s + (53.3 − 92.4i)17-s + (−17.8 − 30.9i)19-s + (37.9 + 13.8i)20-s + (−25.9 − 21.7i)22-s + (85.4 − 31.0i)23-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.469 + 0.171i)4-s + (−0.692 − 0.580i)5-s + (0.407 + 0.148i)7-s + (−0.176 − 0.306i)8-s + (0.319 − 0.553i)10-s + (−0.355 + 0.298i)11-s + (0.319 − 1.81i)13-s + (−0.0532 + 0.302i)14-s + (0.191 − 0.160i)16-s + (0.761 − 1.31i)17-s + (−0.216 − 0.374i)19-s + (0.424 + 0.154i)20-s + (−0.251 − 0.210i)22-s + (0.774 − 0.281i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.11471 - 0.497361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11471 - 0.497361i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (7.73 + 6.49i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-7.54 - 2.74i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (12.9 - 10.8i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-14.9 + 84.8i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-53.3 + 92.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (17.8 + 30.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-85.4 + 31.0i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-16.6 - 94.5i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (191. - 69.5i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-79.7 + 138. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (43.2 - 245. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-229. + 192. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (462. + 168. i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (127. + 107. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (107. + 38.9i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (97.7 - 554. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (124. - 216. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (479. + 830. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-222. - 1.26e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (109. + 621. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (427. + 741. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (241. - 202. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.52837977113856567206136097354, −11.37634228766954648579499771696, −10.18647978152310705942590691794, −8.865012689408389103022183185021, −8.006610244310218289336062923016, −7.20464602665347817395749370280, −5.52686055498244664630390060835, −4.78408099482113427588825290828, −3.17657660594076829778033851696, −0.58084796407474129932162657272,
1.67319380946493503899592333341, 3.42763339124402639336814929811, 4.40572463678925310926009896407, 6.02506380601937038582065664776, 7.39650658080711683951440426648, 8.469117749117547492231797689628, 9.639101902383895362235288163712, 10.90825144427886078555853762454, 11.33769087005983812636876511817, 12.35892733610794844081899435085
