| L(s) = 1 | + (1.64 − 0.548i)3-s + (−0.0121 + 0.0144i)5-s + (0.279 − 0.769i)7-s + (2.39 − 1.80i)9-s + (1.96 − 1.64i)11-s + (−0.388 + 2.20i)13-s + (−0.0120 + 0.0304i)15-s + (−2.75 − 1.59i)17-s + (4.32 − 2.49i)19-s + (0.0383 − 1.41i)21-s + (3.28 − 1.19i)23-s + (0.868 + 4.92i)25-s + (2.95 − 4.27i)27-s + (−5.83 + 1.02i)29-s + (1.49 + 4.10i)31-s + ⋯ |
| L(s) = 1 | + (0.948 − 0.316i)3-s + (−0.00543 + 0.00647i)5-s + (0.105 − 0.290i)7-s + (0.799 − 0.600i)9-s + (0.592 − 0.497i)11-s + (−0.107 + 0.611i)13-s + (−0.00310 + 0.00786i)15-s + (−0.667 − 0.385i)17-s + (0.991 − 0.572i)19-s + (0.00837 − 0.309i)21-s + (0.684 − 0.249i)23-s + (0.173 + 0.984i)25-s + (0.568 − 0.822i)27-s + (−1.08 + 0.191i)29-s + (0.268 + 0.737i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88511 - 0.528486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88511 - 0.528486i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 + 0.548i)T \) |
| good | 5 | \( 1 + (0.0121 - 0.0144i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.279 + 0.769i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.96 + 1.64i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.388 - 2.20i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.75 + 1.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.32 + 2.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.28 + 1.19i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (5.83 - 1.02i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.49 - 4.10i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.45 - 2.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.87 + 1.38i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (3.41 + 4.07i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (5.87 + 2.13i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 0.726iT - 53T^{2} \) |
| 59 | \( 1 + (-6.30 - 5.29i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.02 - 0.736i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.07 - 0.189i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.43 - 12.8i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.99 - 8.65i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (10.3 - 1.82i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.22 + 12.5i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (8.53 - 4.92i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.01 - 1.69i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.20789197223713749382508390792, −9.968588626180564305857815233910, −9.077491874587643712482605609706, −8.549177858853333693423517176872, −7.18761723469835284651591509209, −6.83994265930327468546430433442, −5.21609883558481511396371072619, −3.95512644654116915541048557218, −2.92896833297849168064306761919, −1.41740298688498780633799188779,
1.83555327137093771700385206947, 3.15564560846131366731774563058, 4.23296630024199290989173601489, 5.34805996507061648839341247880, 6.71340819088386266938027272769, 7.73730523956883153965714198623, 8.519818761572468081925269787946, 9.461119101883161946012040229864, 10.08168167688862779939205945009, 11.16640119852809619476521458926
