| L(s) = 1 | + (−0.483 + 1.32i)2-s + (−2.93 − 0.613i)3-s + (−1.53 − 1.28i)4-s + (−7.71 − 1.35i)5-s + (2.23 − 3.60i)6-s + (−0.690 + 0.579i)7-s + (2.44 − 1.41i)8-s + (8.24 + 3.60i)9-s + (5.53 − 9.59i)10-s + (−15.2 + 2.68i)11-s + (3.71 + 4.71i)12-s + (−0.854 + 0.310i)13-s + (−0.435 − 1.19i)14-s + (21.8 + 8.72i)15-s + (0.694 + 3.93i)16-s + (10.6 + 6.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.978 − 0.204i)3-s + (−0.383 − 0.321i)4-s + (−1.54 − 0.271i)5-s + (0.372 − 0.600i)6-s + (−0.0986 + 0.0827i)7-s + (0.306 − 0.176i)8-s + (0.916 + 0.400i)9-s + (0.553 − 0.959i)10-s + (−1.38 + 0.244i)11-s + (0.309 + 0.392i)12-s + (−0.0657 + 0.0239i)13-s + (−0.0311 − 0.0855i)14-s + (1.45 + 0.581i)15-s + (0.0434 + 0.246i)16-s + (0.627 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00182995 - 0.00736161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00182995 - 0.00736161i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 3 | \( 1 + (2.93 + 0.613i)T \) |
| good | 5 | \( 1 + (7.71 + 1.35i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (0.690 - 0.579i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (15.2 - 2.68i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (0.854 - 0.310i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-10.6 - 6.15i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (5.40 + 9.36i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (21.0 - 25.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-19.3 + 53.1i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (37.9 + 31.8i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (17.4 - 30.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (12.2 + 33.6i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-7.20 - 40.8i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (15.9 + 18.9i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 50.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-65.7 - 11.5i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (18.7 - 15.7i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (61.2 - 22.3i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-24.4 - 14.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (10.7 + 18.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (27.2 + 9.90i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (0.0509 - 0.139i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (79.3 - 45.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.4 + 98.9i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
| 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
−15.71713081316890238044754973447, −15.30562485059346714811042749641, −13.34863016213685537678690633352, −12.27532279411946299297429240273, −11.27766965995082418894982355025, −10.00596258588665242886314820144, −8.062467583069944309398557322826, −7.40833568522916347895942264872, −5.71223418392830785362933180869, −4.33309145550125846062191673777,
0.008878731844194105501932087260, 3.55205614885124629113812173392, 5.07347808183806805051262766568, 7.14957062882141058282042341103, 8.345798908613965302197947976017, 10.30531593631983914363450302034, 10.91727637263826496530588210929, 12.07945703781928438147186662095, 12.71731725492399289918372556242, 14.59837766949787657656472169916
