L(s) = 1 | + (−1.80 + 1.04i)2-s + (1.17 − 2.03i)4-s + (1.65 − 2.86i)5-s + 0.717i·8-s + 6.88i·10-s + (−2.30 + 1.33i)11-s + (−2.11 − 1.21i)13-s + (1.59 + 2.76i)16-s − 7.18·17-s − 4.90i·19-s + (−3.87 − 6.70i)20-s + (2.77 − 4.80i)22-s + (4.32 + 2.49i)23-s + (−2.96 − 5.12i)25-s + 5.08·26-s + ⋯ |
L(s) = 1 | + (−1.27 + 0.736i)2-s + (0.586 − 1.01i)4-s + (0.738 − 1.27i)5-s + 0.253i·8-s + 2.17i·10-s + (−0.694 + 0.401i)11-s + (−0.585 − 0.338i)13-s + (0.399 + 0.691i)16-s − 1.74·17-s − 1.12i·19-s + (−0.866 − 1.50i)20-s + (0.591 − 1.02i)22-s + (0.901 + 0.520i)23-s + (−0.592 − 1.02i)25-s + 0.997·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.009462952453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.009462952453\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.80 - 1.04i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.65 + 2.86i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.30 - 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.11 + 1.21i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 + 4.90iT - 19T^{2} \) |
| 23 | \( 1 + (-4.32 - 2.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.50 - 3.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.30 + 1.33i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + (-0.553 + 0.958i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.93 - 5.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.44 - 2.56i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.16 - 7.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.07iT - 71T^{2} \) |
| 73 | \( 1 - 8.01iT - 73T^{2} \) |
| 79 | \( 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.04 - 1.80i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + (-9.47 + 5.46i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.579828843590600014865025006141, −9.061030327933666405836206882993, −8.748801608543519406042264787114, −7.59675118774613298121149984751, −7.10475448713415517119271285216, −6.02171702483192308368103076643, −5.17394172039647490654602645329, −4.41241005241293276897593323442, −2.53198644977878426417449855704, −1.33643750900251354436440273208,
0.00589164195832252731494637731, 1.96648270748726042445413042259, 2.45556501791949226692508881603, 3.52060285533209208798065636595, 5.05554743191158706590785467283, 6.13150237321374550217229269756, 6.97215450159192683683991085092, 7.71538381978128346455944074566, 8.688002747234000358107444366604, 9.359393027279586574670590678667