| L(s) = 1 | + (−0.186 − 2.82i)2-s + (0.730 + 5.08i)3-s + (−7.93 + 1.05i)4-s + (−14.1 − 4.15i)5-s + (14.2 − 3.01i)6-s + (11.2 − 13.0i)7-s + (4.45 + 22.1i)8-s + (0.612 − 0.179i)9-s + (−9.08 + 40.7i)10-s + (15.9 + 24.7i)11-s + (−11.1 − 39.5i)12-s + (−27.8 + 24.1i)13-s + (−38.8 − 29.4i)14-s + (10.7 − 74.9i)15-s + (61.7 − 16.7i)16-s + (61.2 + 27.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.0660 − 0.997i)2-s + (0.140 + 0.978i)3-s + (−0.991 + 0.131i)4-s + (−1.26 − 0.371i)5-s + (0.966 − 0.204i)6-s + (0.609 − 0.703i)7-s + (0.197 + 0.980i)8-s + (0.0226 − 0.00666i)9-s + (−0.287 + 1.28i)10-s + (0.436 + 0.678i)11-s + (−0.268 − 0.950i)12-s + (−0.593 + 0.514i)13-s + (−0.742 − 0.561i)14-s + (0.185 − 1.29i)15-s + (0.965 − 0.261i)16-s + (0.874 + 0.399i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.36259 - 0.225612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36259 - 0.225612i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.186 + 2.82i)T \) |
| 23 | \( 1 + (-74.9 + 80.9i)T \) |
| good | 3 | \( 1 + (-0.730 - 5.08i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (14.1 + 4.15i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-11.2 + 13.0i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-15.9 - 24.7i)T + (-552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (27.8 - 24.1i)T + (312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-61.2 - 27.9i)T + (3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-73.3 + 33.4i)T + (4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-123. - 56.4i)T + (1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-179. - 25.8i)T + (2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-126. + 37.1i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-113. - 33.2i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (194. - 27.9i)T + (7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 - 111. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (165. - 190. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (317. + 366. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-76.8 + 534. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (201. - 313. i)T + (-1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (495. - 770. i)T + (-1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (90.8 + 199. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (138. + 160. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-249. - 849. i)T + (-4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-394. + 56.7i)T + (6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-463. + 1.57e3i)T + (-7.67e5 - 4.93e5i)T^{2} \) |
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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
−11.96162373602328682882221761939, −11.12947299446905162099608659117, −10.17925730842370948403114671155, −9.386089890027320969173570531693, −8.270700731451887916062366909001, −7.29310876434297969474363314879, −4.67031120894900918652519147614, −4.47740619671770722111903104242, −3.27733299154578388423347121373, −1.06760834410243181684665740611,
0.900396343755818169912492485794, 3.25011341276477559137636958518, 4.82599213799191872201412035410, 6.08987586950086034542269815787, 7.40545562029244913746318599035, 7.75278856845501976210941681020, 8.658326637525427266187905647113, 10.04476865551411173871965516161, 11.68452469197050136173370289088, 12.12110823299041557378865838440
