| L(s) = 1 | + (−1.86 − 2.35i)3-s + (7.10 − 1.25i)5-s + (3.36 + 2.82i)7-s + (−2.07 + 8.75i)9-s + (6.85 + 1.20i)11-s + (19.7 + 7.18i)13-s + (−16.1 − 14.3i)15-s + (−21.7 + 12.5i)17-s + (−11.6 + 20.1i)19-s + (0.382 − 13.1i)21-s + (17.3 + 20.6i)23-s + (25.4 − 9.24i)25-s + (24.4 − 11.4i)27-s + (−3.00 − 8.26i)29-s + (23.8 − 19.9i)31-s + ⋯ |
| L(s) = 1 | + (−0.620 − 0.784i)3-s + (1.42 − 0.250i)5-s + (0.480 + 0.403i)7-s + (−0.230 + 0.973i)9-s + (0.623 + 0.109i)11-s + (1.51 + 0.552i)13-s + (−1.07 − 0.959i)15-s + (−1.28 + 0.739i)17-s + (−0.611 + 1.05i)19-s + (0.0182 − 0.627i)21-s + (0.754 + 0.898i)23-s + (1.01 − 0.369i)25-s + (0.906 − 0.422i)27-s + (−0.103 − 0.285i)29-s + (0.767 − 0.644i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.97943 - 0.144049i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97943 - 0.144049i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.86 + 2.35i)T \) |
| good | 5 | \( 1 + (-7.10 + 1.25i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-3.36 - 2.82i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-6.85 - 1.20i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-19.7 - 7.18i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (21.7 - 12.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (11.6 - 20.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-17.3 - 20.6i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (3.00 + 8.26i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-23.8 + 19.9i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (13.9 + 24.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-10.2 + 28.2i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (8.18 - 46.4i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-33.0 + 39.3i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 37.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.7 + 3.47i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-74.5 - 62.5i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (70.1 + 25.5i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-51.3 + 29.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (37.3 - 64.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (90.4 - 32.9i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (42.2 + 115. i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (17.3 + 9.99i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (2.12 - 12.0i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.07912129715965098810078889567, −10.11658710519419972105509019680, −8.935124954823431794553141433331, −8.382380071349352174012854475206, −6.87348755766524321582494192595, −6.09511939142340261947364213223, −5.57248238953664056437436245922, −4.17103458981621404269053928142, −2.06997429626827798262932594610, −1.44165929110931495263120041514,
1.07915935527564723533560135300, 2.79095062025226227726189083708, 4.27657994791579902471768593023, 5.17864940770593910127837374088, 6.30111350306347528325988240874, 6.74245413378715247901073841628, 8.693269060181089756207328546079, 9.130970307308213201968652380323, 10.28273702989761271769289794297, 10.86320484086134324561948638538
