| L(s) = 1 | + (1.36 + 1.06i)3-s + (1.83 + 0.322i)5-s + (−0.441 − 0.526i)7-s + (0.733 + 2.90i)9-s + (−0.0632 − 0.358i)11-s + (2.50 − 0.913i)13-s + (2.15 + 2.38i)15-s + (2.50 + 1.44i)17-s + (−2.50 + 1.44i)19-s + (−0.0430 − 1.18i)21-s + (−1.39 − 1.16i)23-s + (−1.45 − 0.528i)25-s + (−2.09 + 4.75i)27-s + (2.03 − 5.59i)29-s + (−2.94 + 3.50i)31-s + ⋯ |
| L(s) = 1 | + (0.788 + 0.614i)3-s + (0.818 + 0.144i)5-s + (−0.166 − 0.198i)7-s + (0.244 + 0.969i)9-s + (−0.0190 − 0.108i)11-s + (0.696 − 0.253i)13-s + (0.556 + 0.616i)15-s + (0.606 + 0.350i)17-s + (−0.575 + 0.332i)19-s + (−0.00940 − 0.259i)21-s + (−0.289 − 0.243i)23-s + (−0.290 − 0.105i)25-s + (−0.403 + 0.915i)27-s + (0.378 − 1.03i)29-s + (−0.528 + 0.630i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89125 + 0.680144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89125 + 0.680144i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.36 - 1.06i)T \) |
| good | 5 | \( 1 + (-1.83 - 0.322i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.441 + 0.526i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0632 + 0.358i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.50 + 0.913i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.50 - 1.44i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 - 1.44i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.39 + 1.16i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.03 + 5.59i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.94 - 3.50i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.77 + 3.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.29 - 9.05i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (11.8 - 2.08i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.61 + 6.39i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 1.01iT - 53T^{2} \) |
| 59 | \( 1 + (-0.864 + 4.90i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.61 - 8.06i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 4.70i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.29 + 14.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.84 + 11.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.52 + 6.93i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.65 - 1.33i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (12.1 - 6.99i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.204 + 1.16i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.89984970221275967749465935114, −10.20990036750331921526910960451, −9.592623428858447408797075070154, −8.554773009583566490001817619178, −7.84185824141414930390706405608, −6.46325238170004858879191157511, −5.54238556144333439593648551924, −4.23297104908499652495457097813, −3.19719211184383078551979486356, −1.89328802942992250834633740984,
1.48744719604841970377534954391, 2.67657291448391039879149907849, 3.93811116776297122424617237578, 5.50479788393819084527326161567, 6.41407637723119247003069568711, 7.36407609628887263835308761540, 8.428874302253583256117660150691, 9.197730627202892989665314928139, 9.891686924022204665038186922128, 11.04505100854439910501362642013
