| L(s) = 1 | + (−0.347 + 1.96i)2-s + (−3.75 − 1.36i)4-s + (5.66 − 4.75i)5-s + (−11.1 + 4.07i)7-s + (4 − 6.92i)8-s + (7.39 + 12.8i)10-s + (46.3 + 38.8i)11-s + (2.09 + 11.8i)13-s + (−4.13 − 23.4i)14-s + (12.2 + 10.2i)16-s + (52.1 + 90.2i)17-s + (22.8 − 39.5i)19-s + (−27.8 + 10.1i)20-s + (−92.6 + 77.7i)22-s + (13.0 + 4.74i)23-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.469 − 0.171i)4-s + (0.507 − 0.425i)5-s + (−0.604 + 0.219i)7-s + (0.176 − 0.306i)8-s + (0.234 + 0.405i)10-s + (1.26 + 1.06i)11-s + (0.0445 + 0.252i)13-s + (−0.0789 − 0.447i)14-s + (0.191 + 0.160i)16-s + (0.743 + 1.28i)17-s + (0.275 − 0.477i)19-s + (−0.310 + 0.113i)20-s + (−0.897 + 0.753i)22-s + (0.118 + 0.0430i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0376 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.12536 + 1.08377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12536 + 1.08377i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-5.66 + 4.75i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (11.1 - 4.07i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-46.3 - 38.8i)T + (231. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-2.09 - 11.8i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-52.1 - 90.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-22.8 + 39.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-13.0 - 4.74i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (29.4 - 167. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-19.5 - 7.11i)T + (2.28e4 + 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-156. - 270. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (59.6 + 338. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (308. + 259. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-81.8 + 29.7i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 - 753.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (263. - 221. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-669. + 243. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (2.10 + 11.9i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (212. + 368. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-294. + 509. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-167. + 952. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (55.6 - 315. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-49.8 + 86.3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.27e3 + 1.06e3i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−12.73327941359048075321087156186, −11.90192816090157230548087699091, −10.27695175226451752872237377130, −9.419431211368016084524157667593, −8.688961068771452644452190574356, −7.21035938947083084957391985527, −6.32607430750318838366211170792, −5.17408433110416679075933598189, −3.76909015327285870789611303339, −1.52541692996716545481552852729,
0.869830625343945559508013135569, 2.77535084860129078900049051270, 3.88954435899558331636401009119, 5.67097281879717943263057859621, 6.73958273528309452111591623411, 8.184492636713907496738023746114, 9.475884383305116945793972299154, 9.991273383995069615934432891304, 11.27012135260059979547354918797, 11.92943860853421376483481567827
