L(s) = 1 | − 1.38i·2-s + (−2.86 − 0.878i)3-s + 2.08·4-s − 2.23i·5-s + (−1.21 + 3.96i)6-s − 9.90·7-s − 8.42i·8-s + (7.45 + 5.04i)9-s − 3.09·10-s − 3.31i·11-s + (−5.98 − 1.83i)12-s − 17.2·13-s + 13.7i·14-s + (−1.96 + 6.41i)15-s − 3.31·16-s − 1.62i·17-s + ⋯ |
L(s) = 1 | − 0.691i·2-s + (−0.956 − 0.292i)3-s + 0.521·4-s − 0.447i·5-s + (−0.202 + 0.661i)6-s − 1.41·7-s − 1.05i·8-s + (0.828 + 0.560i)9-s − 0.309·10-s − 0.301i·11-s + (−0.498 − 0.152i)12-s − 1.32·13-s + 0.978i·14-s + (−0.131 + 0.427i)15-s − 0.206·16-s − 0.0958i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0815923 + 0.544798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0815923 + 0.544798i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.86 + 0.878i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 11 | \( 1 + 3.31iT \) |
good | 2 | \( 1 + 1.38iT - 4T^{2} \) |
| 7 | \( 1 + 9.90T + 49T^{2} \) |
| 13 | \( 1 + 17.2T + 169T^{2} \) |
| 17 | \( 1 + 1.62iT - 289T^{2} \) |
| 19 | \( 1 + 21.4T + 361T^{2} \) |
| 23 | \( 1 - 35.9iT - 529T^{2} \) |
| 29 | \( 1 + 49.6iT - 841T^{2} \) |
| 31 | \( 1 + 16.0T + 961T^{2} \) |
| 37 | \( 1 - 51.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 79.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 18.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 82.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 81.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 100.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 52.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 34.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 42.7T + 9.40e3T^{2} \) |
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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
−12.04857602363972950377200853583, −11.24680487380412927033904211038, −10.08413563395759773938546211448, −9.539032897020955649989758199669, −7.57395733697367796087943995951, −6.61307846827101813948009877610, −5.65264001111309422265210659911, −3.97032836571623420350817342695, −2.30386020964355399299807213429, −0.34749897484449481012900090182,
2.72242333836543363189667215917, 4.57874158522706496863298537236, 6.01941315263919945998349607305, 6.63506633844276385237880726119, 7.46028032503461463829325069365, 9.226817517852590006901208961281, 10.31245302428692678061500567276, 10.96246260800350688922056781336, 12.34361766898727364827263553039, 12.73015559146077252124206271773