| 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 | \( 1 + (17.4 - 98.9i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−14.59837766949787657656472169916, −12.71731725492399289918372556242, −12.07945703781928438147186662095, −10.91727637263826496530588210929, −10.30531593631983914363450302034, −8.345798908613965302197947976017, −7.14957062882141058282042341103, −5.07347808183806805051262766568, −3.55205614885124629113812173392, −0.008878731844194105501932087260,
4.33309145550125846062191673777, 5.71223418392830785362933180869, 7.40833568522916347895942264872, 8.062467583069944309398557322826, 10.00596258588665242886314820144, 11.27766965995082418894982355025, 12.27532279411946299297429240273, 13.34863016213685537678690633352, 15.30562485059346714811042749641, 15.71713081316890238044754973447
