| L(s) = 1 | + (1.03 + 3.20i)3-s + 0.208·7-s + (12.6 − 9.21i)9-s + (−10.2 − 7.42i)11-s + (−44.2 + 32.1i)13-s + (−27.2 + 83.7i)17-s + (7.21 − 22.2i)19-s + (0.216 + 0.667i)21-s + (−38.7 − 28.1i)23-s + (116. + 84.4i)27-s + (21.7 + 67.0i)29-s + (−72.5 + 223. i)31-s + (13.1 − 40.3i)33-s + (27.7 − 20.1i)37-s + (−148. − 108. i)39-s + ⋯ |
| L(s) = 1 | + (0.200 + 0.615i)3-s + 0.0112·7-s + (0.469 − 0.341i)9-s + (−0.279 − 0.203i)11-s + (−0.943 + 0.685i)13-s + (−0.388 + 1.19i)17-s + (0.0871 − 0.268i)19-s + (0.00225 + 0.00693i)21-s + (−0.351 − 0.255i)23-s + (0.828 + 0.601i)27-s + (0.139 + 0.429i)29-s + (−0.420 + 1.29i)31-s + (0.0692 − 0.213i)33-s + (0.123 − 0.0896i)37-s + (−0.611 − 0.444i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.023358399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023358399\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1.03 - 3.20i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 - 0.208T + 343T^{2} \) |
| 11 | \( 1 + (10.2 + 7.42i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (44.2 - 32.1i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (27.2 - 83.7i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-7.21 + 22.2i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (38.7 + 28.1i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-21.7 - 67.0i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (72.5 - 223. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-27.7 + 20.1i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (172. - 125. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 86.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-42.3 - 130. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (189. + 582. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (623. - 452. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-247. - 179. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (245. - 756. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-184. - 568. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (437. + 317. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (388. + 1.19e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-121. + 374. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-86.5 - 62.8i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-242. - 746. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.66175796154912182406122050844, −10.05845606041862837153062164318, −9.168096000348432563912308951782, −8.395208123925919069645303026446, −7.20312045813883394558037765580, −6.37005643263477562874395785095, −5.02692880833138693187391115206, −4.21592304487139736722834957712, −3.11110344813981915954691939176, −1.64353113484180031851667201887,
0.29357342077850055862235928163, 1.88614206218857065668956117323, 2.90202755608483430887684572897, 4.45110810625383686336985216230, 5.37018877338192877436564925787, 6.63278062144995489703014600663, 7.56021271395824163918386194498, 8.002172592629297067686155271385, 9.385451390728762387403102792830, 10.03236224210697154593583655353
