| L(s) = 1 | + (−2.44 + 1.73i)3-s + (−4.5 + 2.59i)5-s + (3.17 − 5.49i)7-s + (2.99 − 8.48i)9-s + (−8.17 − 4.71i)11-s + (−9.84 − 17.0i)13-s + (6.52 − 14.1i)15-s − 1.90i·17-s − 4.69·19-s + (1.74 + 18.9i)21-s + (−8.17 + 4.71i)23-s + (1 − 1.73i)25-s + (7.34 + 25.9i)27-s + (−2.84 − 1.64i)29-s + (−20.5 − 35.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.816 + 0.577i)3-s + (−0.900 + 0.519i)5-s + (0.453 − 0.785i)7-s + (0.333 − 0.942i)9-s + (−0.743 − 0.429i)11-s + (−0.757 − 1.31i)13-s + (0.434 − 0.943i)15-s − 0.112i·17-s − 0.247·19-s + (0.0832 + 0.903i)21-s + (−0.355 + 0.205i)23-s + (0.0400 − 0.0692i)25-s + (0.272 + 0.962i)27-s + (−0.0982 − 0.0567i)29-s + (−0.662 − 1.14i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.172408 - 0.295463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.172408 - 0.295463i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.44 - 1.73i)T \) |
| good | 5 | \( 1 + (4.5 - 2.59i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.17 + 5.49i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (8.17 + 4.71i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.84 + 17.0i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 1.90iT - 289T^{2} \) |
| 19 | \( 1 + 4.69T + 361T^{2} \) |
| 23 | \( 1 + (8.17 - 4.71i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (2.84 + 1.64i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (20.5 + 35.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (53.5 - 30.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-0.477 + 0.826i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-12.2 - 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (79.2 - 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.4 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-76.1 - 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.32493536890250491468456472547, −11.26398465905026015401742623605, −10.70230545087120722465878286628, −9.801104878953477333529258699936, −8.031322823126616782190067622453, −7.28474345165809796533211691652, −5.75716243159949674768872300727, −4.56958643147551926490471684925, −3.32261068894178789500529427228, −0.23865681734576980419040911121,
2.02762009226977346469245053078, 4.45162547544470042864514415365, 5.35216484770036810621889406606, 6.83878757895171972935704247588, 7.83479549471841972144866279190, 8.851959238502752369320578238840, 10.34581530760745546153594307687, 11.56228005899833088163227017937, 12.09208446929514916313846135371, 12.79828574425227898444252765715
