L(s) = 1 | + 1.89i·2-s + (−2.98 + 0.311i)3-s + 0.398·4-s − 8.25i·5-s + (−0.592 − 5.66i)6-s − 3.60·7-s + 8.34i·8-s + (8.80 − 1.86i)9-s + 15.6·10-s + (−1.18 + 0.124i)12-s − 13.1·13-s − 6.84i·14-s + (2.57 + 24.6i)15-s − 14.2·16-s + 25.6i·17-s + (3.53 + 16.7i)18-s + ⋯ |
L(s) = 1 | + 0.948i·2-s + (−0.994 + 0.103i)3-s + 0.0995·4-s − 1.65i·5-s + (−0.0986 − 0.943i)6-s − 0.515·7-s + 1.04i·8-s + (0.978 − 0.206i)9-s + 1.56·10-s + (−0.0989 + 0.0103i)12-s − 1.00·13-s − 0.488i·14-s + (0.171 + 1.64i)15-s − 0.890·16-s + 1.50i·17-s + (0.196 + 0.928i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0196983 - 0.377819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0196983 - 0.377819i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.98 - 0.311i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.89iT - 4T^{2} \) |
| 5 | \( 1 + 8.25iT - 25T^{2} \) |
| 7 | \( 1 + 3.60T + 49T^{2} \) |
| 13 | \( 1 + 13.1T + 169T^{2} \) |
| 17 | \( 1 - 25.6iT - 289T^{2} \) |
| 19 | \( 1 + 12.5T + 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 9.48iT - 841T^{2} \) |
| 31 | \( 1 + 5.08T + 961T^{2} \) |
| 37 | \( 1 + 34.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 39.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 27.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 107.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 69.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.0T + 9.40e3T^{2} \) |
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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
−11.88135662290270525034445333977, −10.79746098018762677640329885671, −9.739161508581282762800058727241, −8.775564786764045221567098229191, −7.81028547912589273279696339636, −6.79761911332341224163883455327, −5.76869058110615764563927952844, −5.20153584579756052691948379537, −4.12687512460554286170397066925, −1.66195332956059534308134708257,
0.18169977073853167936472318039, 2.26927366034122091926115734906, 3.12870742477057612013613783623, 4.54217151138507666837652525936, 6.12709352637244114194520990895, 6.86970307671540257372834504066, 7.40040662021206509554267525263, 9.523814244530067189941448584677, 10.23918342993471239215827253604, 10.76688062060023489385331850342