| L(s) = 1 | + (−4.5 + 2.59i)3-s + (−40.6 − 23.4i)5-s + (48.9 − 2.45i)7-s + (13.5 − 23.3i)9-s + (−69.3 − 120. i)11-s + 65.6i·13-s + 244.·15-s + (−229. + 132. i)17-s + (385. + 222. i)19-s + (−213. + 138. i)21-s + (−212. + 367. i)23-s + (790. + 1.36e3i)25-s + 140. i·27-s + 580.·29-s + (−1.45e3 + 841. i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (−1.62 − 0.939i)5-s + (0.998 − 0.0500i)7-s + (0.166 − 0.288i)9-s + (−0.573 − 0.992i)11-s + 0.388i·13-s + 1.08·15-s + (−0.794 + 0.458i)17-s + (1.06 + 0.615i)19-s + (−0.484 + 0.313i)21-s + (−0.400 + 0.694i)23-s + (1.26 + 2.19i)25-s + 0.192i·27-s + 0.690·29-s + (−1.51 + 0.875i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0766 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0766 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6533536544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6533536544\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 + (-48.9 + 2.45i)T \) |
| good | 5 | \( 1 + (40.6 + 23.4i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (69.3 + 120. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 65.6iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (229. - 132. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-385. - 222. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (212. - 367. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 580.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (1.45e3 - 841. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-762. + 1.32e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 2.59e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.59e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-2.48e3 - 1.43e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-1.12e3 - 1.94e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-416. + 240. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-4.00e3 - 2.31e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.86e3 + 4.96e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 3.64e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (4.73e3 - 2.73e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-479. + 830. i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 7.57e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (3.49e3 + 2.01e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 3.80e3iT - 8.85e7T^{2} \) |
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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
−12.04305325220305114308957902551, −11.43853508363103375315769792029, −10.74239506908866910300917700324, −9.053488081145192085087056283857, −8.197624798806730086660236155521, −7.41855901841371430064327387837, −5.59034339248524231195365424702, −4.62346367380299761886186652300, −3.63918120160600938575683516820, −1.11693741062495196629049120727,
0.31780344569406581140370555419, 2.47822717176569655041265974609, 4.11662444042038681139750090310, 5.15900658828126588479182342450, 6.98051625312892724966199097623, 7.48154255575738114765922878770, 8.445725002113900664770002693255, 10.21905187750055331859076039775, 11.15350572899736059351551327898, 11.67476161946339768676562565059
